GIFT OF
ASTRONOMY
LIBRARY
TABLES OF MINOR PLANETS DISCOVERED BY
JAMES C. WATSON
PART II
ON v. ZEIPEL'S THEORY OF THE PERTURBATIONS
OF THE MINOR PLANETS OF THE HECUBA GROUP
MEMOIRS
or THK
NATIONAL ACADEMY OF SCIENCES
THIRD MEMOIR
WASHINGTON
GOVEBNMENT PRINTING OFFICE
1922
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ASTRONOMY
LIBRARY
NATIONAL ACADEMY OF SCIENCES.
Volume XIV.
THIRD MEMOIR.
TABLES OF MINOR PLANETS DISCOVERED BY
JAMES C. WATSON.
PARTIL
ON v. ZEIPEL'S THEORY OP THE PERTURBATIONS OP THE
MINOR PLANETS OF THE HECUBA GROUP.
BY
ARMIN O. LEUSCHNER, ANNA ESTELLE CLANCY, AKD
" SOPHIA H. LEVY.
50f>877
.VI /C 'Hi i ii foV
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CONTENTS.
Page.
Preface 7
Introduction g
I. Formulae and tables for the Hecuba group, according to the theory of Bohlin-v. Zeipel. and an example of their
se 10
Determination of constant elements and of perturbations of the mean anomaly 10
Perturbations of the radius vector 20
Perturbations of the third coordinate 21
Check computation 22
Computation of the perturbations for the time t 22
Comparison of the revised with v. Zeipel's original tables 27
Table A 28
Table B 30
Table C 31
Table D 34
Table E, 35
Table E 2 35
Table F 36
Table G 38
II. Tables for the determination of the perturbations of the Hecuba group of minor planets 41
Development of the differential equations for Wand for the third coordinate 41
Integration of the differential equation for W. 78
Comparison of tables 120
Perturbations of the mean anomaly 121
Comparison of tables 134
Perturbations of the radius vector 137
Perturbations of the third coordinate 140
Comparison of tables 146
Constants of integration in nSz and v 146
Comparison of tables 155
Erata in " Angenaherte Jupiter-Storungen fur die flecufco-Gruppe," H. v. Zeipel 156
Erata in ' ' Sur le Developpement des Perturbations Planetaires, " 1-7 and Tables I-XX, Karl Bohlin 157
5
!*T II
V -all \0 i'uutiq vflts >.)}
PREFACE.
Part I of "Tables of Minor Planets Discovered by James C. Watson," containing tables
for 12 of the 22 Watson planets, was published in 1910 in the Memoirs of the National Academy
of Sciences, Volume X, Seventh Memoir, with a preface by Simon Newcomb, in which he
gives an account of the early history of the investigations of the perturbations of the Watson
planets under the auspices of the Board of Trustees of the Watson Fund.
In the introduction to Part 1 1 reference is made to the Watson planets of the Hecuba
group, for which it was found necessary to construct special tables on the plan of Bohlin's
tables for the group 1/3. A comparison of these tables with similar tables by v. Zeipel remained
to be made before applying either of them to the development of perturbations of planets
of the Hecuba group. This comparison was completed in 1913 with the assistance of Miss A.
Estelle Glancy and Miss Sophia H. Levy, with the results set forth in the following pages.
Publication of these results was delayed, partly because it seemed desirable to verify the
tables by application to a number of planets and partly on account of interruptions caused in
recent years by war conditions. Miss Glancy, in particular, had undertaken to test the accuracy
of our tables, which we had applied to v. Zeipel's example, (10) Hygiea, by further investi-
gations on this example after joining the Observatorio Nacional at C6rdoba in 1913. This
test has now been completed with highly satisfactory results. The tables have also been
successfully applied to the Watson planets of the Hecuba group, including (175) Andromache,
which, on account of unusually large perturbations and other unfavorable conditions, forms
so far the most striking example of the applicability of the Bohlin-v. Zeipel method and of our
revised tables for the Hecuba group.
The plan of work included conferences, in which Miss Glancy and Miss Levy took a leading
part, for the discussion of the Bohlin-v. Zeipel method, involving verification of all mathe-
matical developments and formulation of plans for the construction of tables, and, after the
appearance of v. Zeipel's tables, for the comparison of v. Zeipel's original, and OUT revised tables.
The numerical work was carried out by Miss Glancy and Miss Levy, who have also contributed
very largely to the theoretical part of the work, and have prepared the principal details of the
manuscript.
To avoid confusion v. Zeipel's notation and method of procedure have been followed
throughout in completing our tables for the Hecuba group, which were well under way when
v. Zeipel's memoir appeared.
To aid computers in the use of the formulae and of the revised tables, Miss Glancy has
prepared detailed directions illustrated by an application to (10) Hygiea, the example first
chosen by v. Zeipel. These are contained in the first section of the present memoir.
Miss Glancy's contributions to this investigation and her work on (10) Hygiea were accepted
by the University of California in partial fulfillment of the requirements for the degree of
doctor of philosophy.
Miss Levy's contributions and her work on (175) Andromache were similarly accepted for
the same degree.
It seems highly desirable to make the tables for the development of the perturbations of
minor planets of the Hecuba group at once available to astronomers. They are therefore
published herewith, in advance of the perturbations and tables of the remaining Watson planets,
as Part II of "Tables of Minor Planets Discovered by James C. Watson." One or two parts,
which are to follow, will contain all the numerical results for the perturbations and tables of
Watson planets not published in Part I (1910).
This memoir is presented in two sections. The first section, entitled "Formulae and Tables
for the Hecuba Group, according to the Theory of Bohhn-v. Zeipel, and an Example of their
Pp. 200-201.
8
PREFACE.
[Voi. xiv.
Use," contains a collection of the formulae to be used for any planet of the Hecuba group, the
general tables of the perturbations which must be employed, and a more complete application
of the formulae and the revised tables to the plane* (10) Hygiea, than v. Zeipel gives. The
second and more extensive section, entitled "Tables for the Determination of the Perturbations
of the Hecuba Group of Minor Planets," concerns the construction of the tables and their dis-
cussion with reference to the corresponding tables by v. Zeipel. It forms the preliminary part
of the in restigation but is presented last as supplementary to the final results given in the
first section.
In the second section the tabular values which differ from the corresponding numbers in
v. Zeipel's tables are placed in brackets. The general Tables XXXV, XXXVIII, XLIII,
LIV, LVi, LVn, LVI, LVII, of the second section, which, in order, are required to compute
the perturbations of any planet of the Hecuba group, are repeated without brackets at the
end of the first section as Tables A, B, C, D, E 1; E 2 , F, G, so that the first section is complete
in itself for use in developing the perturbations of any planet of this group without the necessity
of reference to the second section.
A general account of the investigations of the perturbations of the Watson planets was
presented to the Academy on April 16, 1916, and is published in the "Proceedings of the
National Academy of Sciences," Volume 4, No. 12, March, 1919.
' ARMIN O LEUSCHNER.
-;.' WASHINGTON, D. C., 1918, December.
fun
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TABLES OF MINOR PLANETS DISCOVERED BY JAMES C. WATSON.
By AHUM O. LKUSOTNZR, ANNA ESTELLB GLANCT, AND SOPHIA H. LETT.
*
.
INTRODUCTION.
Those planets whose mean daily motions are approximately 600" are classed with the
planet Hecuba, or, in the group for which
u= = K1-0
n 2
where n' and n are the mean daily motions of Jupiter and the planet, respectively, and w is a
small quantity.
Among the minor planets discovered by James C. Watson there are several of this type.
In the course of the general program of determining the perturbations of the Watson asteroids,
there arose the necessity of computing special tables for the Hecuba group in preparation for
the application of Bohlin's method to individual planets.
General tables for the group $ were in the process of construction, under the direction of
Professor Leuschner, 1 according to the method of Bohlin,* when tables for this group were
published by H. v. Zeipel.* The computers, Dr. Sidney D. Townley and Miss Adelaide M. Hobe,
made a comparison of their tables with those of v. Zeipel and found certain discrepancies
Because of this fact the completion of the tables for the Hecuba group was deferred. These
discrepancies have been explained, as a result of a careful investigation, and the tables have
been completed by Miss A. Estelle Glancy and Miss Sophia H. Levy, under the direction of
Professor Leuschner.
In the completion of the tables, v. Zeipel's method and order of procedure have gener-
ally been followed. There are numerous discrepancies between our tables and v. Zeipel's. As
far as possible, with the aid of the original manuscript, kindly forwarded by the author, we have
traced the source of these disagreements. In some of the more complicated functions it was
not possible to do so, and these discrepancies remain unexplained. Our own results, however,
are substantiated by the employment of independent developments. Further, where we found
terms omitted which were of the same order as those which were included, we frequently
extended the tables. In this connection, it is pertinent to remark that it is very difficult to set
up a consistent criterion for the omission of terms. With the exception of a few scattered
negligible terms, our tables are published in full. They contain terms which may ordinarily be
omitted, yet their numerical magnitudes depend upon the elements of the particular planet
under consideration, and their use is left to the computer's judgment. Many of them are
incomplete, i. e., the tabulated coefficients do not necessarily include all the terms of a given
degree in the eccentricities or mutual inclination or of the small quantity w, which depends
upon the difference between the planet's and twice Jupiter's mean motion. In other words, the
coefficients may not contain all the terms of a given degree having the factors
W, Jf, ,', ft
which are defined on page 12. But, assuming certain numerical limits for the fundamental
auxiliary functions, the coefficients are of this magnitude. The value of the additional terms
will be shown best in an application of our tables to the same planet for which v. Zeipel computed
the perturbations.
Unless stated otherwise, the references to Bohlin refer to the French edition and are
designated by B. References to v. Zeipel are designated by Z.
1 Memoirs of the National Academy of Sciences, Vol. X, Seventh Memoir, p. 200.
Fonneln und Tafeln rur gruppenweisen Berechnung der allgemeinen StSrungen benachbarter Planeten (Tpsala, 1896).
Sur le DeYetoppement des Perturbations Plangtaires (Stockholm, 1902).
1 Angenaherte Jupiterstorungen fflr die Hecuba-Gruppe (St. Pfitersbourg, 1902).
9
tw
I. FORMULAE AND TABLES FOR THE HECUBA GROUP, ACCORDING TO THE
THEORY OF BOHLIN-v. ZEIPEL, AND AN EXAMPLE OF THEIR USE.
DETERMINATION OF CONSTANT ELEMENTS AND OF PERTURBATIONS OF THE MEAN ANOMALY.
The planet (10) Hygiea was selected by v. Zeipel as an example of the use of his tables for
the group . We have used it as a preliminary example for the application of our own tables,
so as to provide further comparison of our tables with those of v. Zeipel.
This example is presented with the direct purpose of meeting the needs of the computer.
For this reason, no attempt is made to explain the significance of the functions involved, yet
their use will be less mechanical, if, in a general way, some of the essential principles under-
lying their development are understood. The theory of v. Zeipel is taken up in the second
section of this memoir.
The method proposed by v. Zeipel is a practical adaptation of Bohlin's method of com-
puting the perturbations by Jupiter upon planets whose mean motions bear nearly commen-
surable ratios to that of Jupiter. In particular, the formulae are derived for the planets of the
Hecuba group. Tracing the history of this method one step further back, Bohlin's method is a
modification of the theory of Hansen for the indeterminate case of nearly commensurable mean
motions. Or, concisely, in v. Zeipel's own words, "Die benutzte Methode kann einfach dadurch
charakterisirt werden, dass die Differentialgleichungen von Hansen mittels des Integrations-
verfahrens des Herrn K. Bohlin gelost worden sind." 1
Certain principles of Hansen are fundamental to an understanding of some of the important
equations. Briefly, the perturbations are reckoned in the plane of the orbit and perpendicular
to it. In the plane of the orbit n5z signifies the displacement in the planet's mean anomaly
(8z is the perturbation in the time) ; v gives the disturbed radius vector through the relation
u
and the displacement in the third coordinate is denoted by =. With Hanson's choice of ideal
COS v
coordinates, the fundamental analytical relations are:
t nl
s e sin s = nt + c + ndz
rcosf=a (cos e -) ft)
fcH!?i /r 3_:_
> in/_ * in
-
s<( vlhji'
j'jn
oiii ni'iftl 1-J vuul . I.HP
^ = coin s * n *"
Jz=o"/cosa (2)
Jv=/8co b
"<*' / Jo
Az = dp cos c
x = r sin a sin (A' +/) +Ax
y = r sin b sin (B' +/) + Ay (3)
z = r sin c sin (C' +/) + Az
where s,f, f are fictitiously disturbed coordinates, which, in connection with constant elements
and the perturbations n5z, v, and = give the true position of the body. A', B', C', sin a, sin b,
COS 1
sin c are the constants for the equator. The notation for the eccentric anomaly and the true
anomaly is v. Zeipel's; in Hansen's notation they would be written e,f.
' Angenaherte Jupiterst8rungen fur die Hecuba-Gruppe, p. I.
10
NO. 8.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 11
When Jupiter's mean motion and that of the planet are nearly commensurable, the inte-
gration of Hansen's differential equations becomes impracticable through the presence of large
integrating factors. The integrals are of the form:
1 sinf 1 oo <i< + 00
\(in-i'n')t\ A ^.,^
cos 0<t'<+oo
./. i'n'Y
71*1 1 I
V n /
For the Hecuba group the mean motion is approximately twice the mean motion of Jupiter.
Hence, for exact commensurability,
n2' V. iVV
nH % --
V n )
By introducing the exponential in place of the sine and cosine, the indeterminateness can
be removed, for if in i'n' = Q, then V-Km-i'')<_ j This is one of Bohlin's modifications.
For any given planet the ratio is not exactly commensurable, and the developments are
originally made for the case of exact commensurability. They are then expressed, for a given
case, by Taylor's series in ascending powers of a small quantity w, which depends upon the
difference between the real ratio and exact commensurability. In addition to positive powers
of w there will occur negative powers. They are due to the following causes. An argument 6 is
introduced (see p. 13), from which the mean anomaly of Jupiter is eliminated through the intro-
duction of w. It is a necessary consequence of the form of the partial differential equations in
j r\
which -r appears, that the integration of first-order terms shall contain vr l and that higher
order terms shall contain other negative powers. Hence the integrals are series in both posi-
tive and negative powers of w.
In distinction to the method of Hansen the elements appear explicitly in the arguments
or as factors in the terms of the series.
An important feature of v. Zeipel's theory is his treatment of the constants of integration.
Since the method is essentially Hansen's, the constants of integration must be determined con-
sistently with that method. Given osculating elements, the constants of integration are deter-
mined by the condition that, at the date of osculation, (t = 0), the perturbations and their
velocities shall be equal to zero.
v. Zeipel adopts osculating elements as his initial elements. With these elements and the
perturbations and their velocities at the date of osculation, he computes elements, designated
by the subscript unity, in which the constants of integration are absorbed. They are analogous
to Hansen's constant elements, i. e., the fundamental equations of Hansen are valid.
Our transformations of the elements differ from v. Zeipel's in two respects. First, the
constants in - ., and in its velocitv have not been introduced into the elements i, Q, but
cos ^
are treated in the usual Hansen manner. Second, v. Zeipel introduces certain terms in the
perturbations which have the same period as the planet (argument s), into the elements to
form mean elements. This has not been done.
The general tables, XXXV, XXXVHI, XLIII, LIV, LVi, LVn, LVI, LVTT, which are
required in computing the perturbations, are given at the conclusion of the formulae. The
formulae for any planet of the group are given completely, and they are supplemented by
numerical values for the planet (10) Hygiea.
The references to v. Zeipel's paper are indicated briefly by Z, followed by the number of
the page.
The osculating elements of the planet are taken from Z 139; the elements for Jupiter are
taken from Astronomical Papers of the United States Xautical Almanac Office, Vol. VII, p. 23.
12
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
tvoi.xiv.
(10) Hygiea.
Epoch, 1851, Sept. 17.0, Ber. M. T.
OSCULATING ELEMENTS.
7i = 634?850 = 0? 176347
*> = 546/28= 5?7713
;r = 227 46.61=227.7768
ft = 287 37.19 = 287.6198
= 300 9.42=300.1570
i n = 3 47.14= 3.7857
Jupiter.
Epoch, 1851, Sept. 17.0, Ber. M. T.
'illl ',' ', r. -\\ L --in:
MEAN ELEMENTS.
n'= 299?1284= 0?0830912
<p' = 245/95= 2?7658
JT'~ 11 54.45= 11.9075
ft'= 98 55.97= 98.9328
' = 272 58.48 = 272.9747
i'= 1 18.70= 1.3117
to*-!
. o
. C = 126 59.81 = 126. 9968 c' = 199 57.70= 199. 9617
Mean equinox and ecliptic, 1850.0.
Epoch, 1851, Sept. 16.96279 Gr. M. T.
The following notes in regard to these elements are of importance:
Jupiter's elements were first taken from Z 139. They were used only in the equations
numbered (1). In these equations either set of elements may be used with sufficient accuracy.
In fact, it is not necessary to know Jupiter's elements as accurately as those of the planet, for
they appear only in the arguments of the perturbations. We have adopted Hill's values of
the elements and Newcomb's value of the mass of Jupiter. The tables of the perturbations
are based, however, on Bessel's value for m'. To correct the perturbations for the improved
value, it is only necessary to multiply them by 1.0005, and this is done in the formulae which
follow.
The original epoch of Jupiter's elements was 1850.0 Gr. M. T. It was changed by the
formula c' = 148 1/97 + '* (4)
'<}<; iv (!.; <
The elements of Hygiea are very good osculating elements, computed by Zech. They
include perturbations by Jupiter, Saturn, and Mars and are based on five oppositions. The
reference for these elements is doubtful, for in Astronomische Nachrichten 39, 347, the elements
given by Zech are not identically the same, although the differences are very small. The values
given by v. Zeipel were probably taken from Zech's manuscript, to which he had access. They
may, therefore, contain some later corrections.
The auxiliary quantities ^, *, J are first computed by the formulae:
sin ^ J sin ^ (*+*)=sin^ (ft - ft') sin i (i +*')
sin ^ J cos g (^+*)=cos ^ (ft - ft') sin 2 (*o~*')
cos K J sin o (*-*)= sin ^ (ft -ft') cos 7, (i +i')
(5)
Then follow
cos sr J cos K ( 1 i r -*)=
sin Sfr sin $ sin
, &') cos o (*o~ *')
sin i sin i'
- cos
cos 2
s-
J =n -n' ; j- =
cos
NO. a.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 13
and the arguments for the date of osculation:
.nwbl-M
o = Lr - g' where g' = f ' + [n'fel ; [n'te'J = (9.5215) sin. 1 15?326, (7)
where the coefficient in parentheses is logarithmic in degrees.
1
e e sme, = c :r = ^e <> + + J lt (8)
(10) Hygiea.
* = 186?4792 n, = 302?3984 J. = 215?8679
* = 357. 7586 n' = 86.5305 J.= 28.9289
J= 5.0856
log ,= 8.70139 log /* = 7.29275 0, = 223.2334 (a)
log i)' = 8.38238 log i = 8.94739 ^ e ~ ( c ' + f 72 '^^ = 223 -2445 (6)
= 8.76072 \ c,-c' = 223.5448 (c)
See footnote. 1
e = 131?3236; r=145?0746
With these initial quantities all the arguments and factors in Table L^*I or F are computed.
The required function, w w t , is computed by successive approximations, the first approximation
being
Wg
In the first trial the smallest terms and the last digit may be omitted; the second trial should
be accurate; a third trial, if necessary, will require only corrections to the largest terms.
The mean motion n is then given by
In'
n== l^i>
honinmlob -i ., Td bsloiwb 0=-- \ -..u) - * -.ui.,,>. badiuJ.ii. 7fcooiJii-.il 9ii
-noc Jam id* jbuotuj
(10) flj/yiec,
_ ,, . . , ,
The three successive trials for u> give
W -w.
+ 0.00388 w = + 0.06 1 208
+ 0.003541 logw= 8.78681
+ 0.003568 n = 637?2633
.
Designating by C and S series to be computed next from Table LVII or G, it is evident by
inspection of Table LYII that
C'cos i^ + S sin <}> = Ic cos (i{> + X)=Ic cos X cos <pZc sin X sin $
from which
C= Ic cos X; S=-Ic sin X (10)
> Three numerical values for the argument i, are given. According to the theory (see footnote, Part 2, p. 147), (a) is rigid; (4) is rigid within the
accuracy of the developments by v. Zeipel; (e) is an approximation which v. Zcipel used and which is used here. The value (i) is preferable.
Inequation (6), [n'Sz 1 ] +0*.31U and is tho complete perturbation of Jupiter by Saturn taken from Hill; in all other parts of the computation
n'li'} is only the long period term used by v. Zeipel.
14
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
To make the order of computation evident, the successive steps for a group of terms for
Hygiea are given.
X
c
X
-s
+ C
-5r+60 +6J
+ 0.4
* '
111. 10
+ 0.4
ft
- 0.1
-4r+60 +64,
+ 1.9
256. 18
- 1.8
- 0.5
-3r+60 +6J
+ 4.6
41.25
+ 3.0
+ 3.5
-2r+69 + 64,
+ 6.8
186. 33
- 0.7
-6.8
- r+60 +64,
+21.5
331. 40
-10.3
+18.8
60 +6J
-63.0
116. 48
-56.4
+28.1
r+60 +6J
- 4.0
261. 55
+ 4.0
+ 0.6
2r+60 +64
- 3.1
46.63
-2.2
- 2.1
3r+60 +6J
- 1.9
191. 70
+ 0.4
+ 1.9
The second column contains the sum of the numerical coefficients multiplied by their respec-
tive factors i&ipy'vf 1 \ The columns S and + C con tain the required terms from this group in
the table. They can be computed at the same time if a traverse table is used. 1
From S and C the elements n and <p can be computed by the formulae :
i
e sin (n 7r )=S cos <p u
e cos (n TT O ) = e + C cos V
e = sin <p
In place of ij a , J , - the following are used hereafter:
e
-TOOT-*
(ID
(ff TO)
.dj
(12)
(70)
S=+1215?0
<7= +2191.1
A = 218?8882
;:= +3?0203
r = 230. 7971
S= 31?9492
log, = 8.7451 7
IB hi 1-ift >r!t rtl
(I ft I'.KtBIU'i'MI 9<i
(io(, bdT
There remains one more element to determine, namely, c, but the computation must be
deferred until we know the perturbation nSz at t = 0. (See equation (1), page 10 or page 16.)
The fictitiously disturbed eccentric anomaly at the time t = denoted by is determined
through the relations :
(13)
s n e n sin
where is calculated with the aid of Astrand's table 2 ;
iA rlivilj *tVMSi9*V
(14)
v<i
131?3236
li.Oio TIV.I 'iki
(10) Hygiea.
! = 127?6064
i-csine, = 122?5578
The perturbation nSz is computed as follows :
The function 1 + 0(0) is computed from Table XXXVIII or B. The coefficients are mul-
tiplied by their respective factors, the trigonometric functions of the arguments are expanded,
and the coefficients of
5111
are collected, (j is the numerical coefficient of t?) .
1 Memoirs of the National Academy of Sciences, Vol. X, Seventh Memoir, p. 218.
Hulfstafeln zur leichten und genauen Auflosung des Kepler'schen Problems (Leipzig, 1890).
Ko.3.)
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
15
(10) Hygiea.
1 + 0(0) = (1 -0.008064) {1 -0.055937 sin 20 + 0.017170 cos 20
+ 0.016057 sin 40 + 0.012244 cos 40
+ 0.000905 sin 60-0.005081 cos 60+ .....
+ (*-*.)( + 0.000007 -0.000490 sin 20-0.001266 cos 20
-0.000361 sin 40 + 0.000409 cos 40+ ..... )}
where the coefficients are in radians, and is the value of at t = 0.
-j ji iwjilq *Jii) Wi .-f.R ' ,-. PI imirfJiiHSpI 'y:ii ,j, xitulv, ;.,
Let 1 + a be the nontrigonometrical term in 1 + 0(0), take it out as a common factor, and
denote the numerical coefficients by A 2 , B v A v B v A t , B s , b a , a 2 , b 2 , a t , b t , respectively.
With these coefficients the following are computed :
K= T^w sin 1"
(15)
r>
C.
Ift'.V I'..
There are check formulae for these quantities in Z 134, equation (153), (161')- In equa-
tion (153) there is a misprint; in equation (161') there are two misprints. The errors and their
corrections are noted in the list of errata which accompanies the second section of this paper.
A part of the long period terms in ndz, denoted by [ruJz],, is expressed by
sn
cos
sn
cos
sn
cos
^^
(10) Hygiea.
1+0(0) = (1-0.008064) {1-0.056384 sin 20 + 0.017308 cos 20 + 0.016186 sin 40
+ 0.012342 cos 40 + 0.000912 sin 60-0.005122 cos 60+ . . .
+ (0-0,) ( + 0.000007 -0.000494 sin 20-0.001276 cos 20-0.000364 sin 40
+ 0.000412 cos 40+
A, = + 0.01 7308
Bj = - 0.056384
AI= +0.012342
B,= +0.016186
A t =- 0.005 122
=+ 0.000912
'
. . )+....}
6, = +0.000007
a, = - 0.000494
6, = - 0.001276
a 4 = -0.000364
6 4 = +0.000412
\ Ho'Ji nc:ij'u:j'
16 MEMOIKS NATIONAL ACADEMY OF SCIENCES. [VOLXIV.
Unit of A 2 , etc., is one radian
[<H= (3.59592) sin 2 +^(C~Co) [(0.933J sin 2
+ (4.09785) cos 2 + (0.521) cos 2
+ (3.0783) sin4 +(0.085) sin 4^
+ (3.2230J cos 4^ + (0.005) cos 4
+ (2.4390.) sin 6C + ...... ]
+ (1.494,,) cos 6C + v ,w**i (!
in which the coefficients are logarithmic in seconds of arc. For this planet it is not necessary
to include C ".
uu mis t
In equation (16) let
S n = fc cos K C n = sin A .
S' n = -fc' sin J5T' C",-*' cos JT
Then
cos (c+^')+ Jv t u.-, (18)
The argument C ia given by the relation:
(51)
and ^ is the value of at 2 = 0, in which, [w'fe'J, the long period term between Jupiter and
Saturn is:
(9.5215) sin{ (9.58539) T+ 1 15?326}, R) (20)
where the numerical coefficients are logarithmic in degrees, and Tis measured from the date of
osculation in Julian years.
The complete expression for the long period term in ndz is :
;?.!) - 2 o-KAS + B^/w ,,\ wriT
- [ndsll+ l-wl + $(A 2 2 +B 2 2 )\2 s [n ' zl j
It is important to remark that, in equations (19), (21), the eccentric anomaly is computed
by the usual formula,
' JO 4- } iu* s ~ e sin $ = c + nt + n8z : wo ?> - Jata] C 1 )
in which the multiples of 2^ must be retained, for is used here as if it were the time. Since
ndz is unknown, the computation is by successive approximations.
(10) Hygiea.
[7^3],= (4.1 1837) sin (2+ 72?5246)
+(3.3130) sin (4C + 305. 627)
+ (2.442) sin (6C + 186.48)
-.o'jpV-'JUO.O ..... i'lH )( iJ)jl).-.TO
[ +|(C-Co){(0-963)cos(2C+ 68.83)
+ (0.199) cos (4C+309.75)+ ....}+ ....
in which the coefficients are logarithmic in seconds of arc.
lQ g i
The argument & in (ndz [ndz]), the short period part of 71^2, is given by
C (22)
and the function itself is computed from Table XXXV or A.
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 17
The numerical coefficients in Table XXXV or A are multiplied by their respective factors
and the terms are then collected in the form
r^z-[ndz] = lC S ^(i^e+j^ + U-H) (23)
By expanding the trigonometric functions, the known part of the argument, namely,
IcA-lZ
is incorporated in the coefficients, and the terms are collected in the form :
ndz - [ndz] = la sin x + 2b cos
where
x=2
Let
a = cos K
a' =- sin K'
a"= Jc" cos K" b" = Jc"smK"
Then
ndz - [ndz] = Ik sin Gt+Z) + (*-*,) SV cos (x+ K' )
>-tfo
mo
' sinx + -^' cos x)
" sin x + 2b" cos x)
(24)
6
b'
=t
=&'
sin K
cos 1C'
(25)
(26)
The tabulation of ndz [ndz] for (10) Hygiea is given on page 27.
Finally, the complete perturbation in the mean anomaly is:
8 ndz = [ndz]+(ndz-[ndz]) (28)
It is now possible to determine c by successive approximations from equations (20), (19),
(18), (21), (22), (27), (28).
From equation (1), which holds for any time t,
c=ie sin ^ ndz
t = o (29)
= i
As a first approximation
ndz = c = e, e sin s l
Introducing this value of c in equation (19), a first approximation for ndz is made. For <=0,
C-Co)=0 (30)
(*-0 8 ) =0
Substituting the value of ndz in equation (29), and computing a new value of c, the process of
solution by trials is repeated until a satisfactory agreement is reached.
110379 22 2
-K?;J
18
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
(10) Hygiea.
Below is the last approximation for the constant c.
(See tabulation of mz \n3z\ on pg. 27.)
L'l.
24**
x+K
log sin (*+ A')
, Si n (x+ .)
Approx. ndz
+0?6124
[ sin ,
122. 5578
i*+ &
285?683
323?619
9. 7732 n
- 282"
Approx. c, equ. (1), p. 10
1 w 10
-g- c > P- 13
121. 9454
57. 240
frS!
9.443
93. 203
291. 021
258. 21
9. 9701 B
9. 991 n
- 680
- 260
*H2! r r 7 n 1
217. 278
lff+30
158. 077
241. 837
183.00
335. 37
8. 719 n
9. 620 n
- 6
- 17
o C (, , p. -L*
127. 606
135. 14
9.848
+ 25"
I.UP-W
+3. 9053
+20
211. 366
295. 126
288. 414
256.179
9. 9772 n
9. 9872 n
-3403
- 723
[n'oV], equ. (20), p. 16
+0. 3003
+ 60
18. 886
223. 38
9. 837 n
- 168
(9.99572)(|f 1 -K<?2'])
+3. 5697
+80
102. 646
316. 154
186.8
14.11
9. 073 n
9. 387
- 5
+ 23
- +40
39. 914
129. 91
9.885
+ 3
( \
137. 049
74.51
9.984
+ 121
5 i [n'dz'] j
-0. 0752
ff+50
220. 809
39.53
9.804
+ 44
'
304. 569
3.25
8.754
+ 1
f, equ. (19), p. 16
220. 848
81. 696
2f+40
338. 972
62. 732
236. 180
209. 15
9. 9195 n
9. 688 n
- 80
- 19
,
163. 392
2e+60
146. 492
183.0
8. 72 n
- 1
fir
245. 088
s +50
348. 415
2.4
8.62
tt+n
72. 175
327.9
9.72 n
- 4
2+ 72?525, p. 16
4r+305. 627
154. 221
109. 019
+ 217"-5654"
6r+186. 48
71.57
ndz [noz]
/ - 5437"
{- 1?5103
log sin
log sin
log sin
9. 6384
9. 9756
9. 9771
(8.3192 B )(| 1 -KoV])
[nM
7102, equ. (21)
- 0.0752
+ 2. 1994
+ 0.6139
+ 5712"
C=C 1
121. 9439
+ 1944
+ 262
(6. 8050 B V ,
^ 1 ! 1
- 0.0778
me* 'i i
/ +7918"
c.,, p. 19
. ODD!
(9. 67154)^2],
\ +2? 1994
+1. 032
l-w
+57. 240
2 C>
lw
0, equ. (22), p. 16
221?880
217. 278
2 q c f
20
83.760
305. 640
(9. 6715)1^2],
+ 1.032
.- - ; -.'
, equ. (22)
221. 880
40
167. 520
50
29.400
60
251. 280
70
113. 160
80
335. 040
Jf , p. 14
63.803
f
127. 606
t,
191. 409
i i I If tl) I
2f
255. 212
319. 015
i
Collecting the elements, and adopting a change of notation, introduced at this point by
v. Zeipel, namely, the addition of the subscript unity to the elements just now determined,
n, = 637?2633 = 0? 17701 758
<?,= 6? 3858
^ = 230.7971
c, = 121. 9439
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 19
These elements are constants; they differ from constant osculating elements only by the
constants of integration in ndz and v. They are to be used in the same manner as Hansen uses
constant osculating elements.
It is possible, in a similar manner, to absorb the constants of integration in the third coordi-
nate in the elements i and &, but this transformation will be omitted.
It is a convenience to the computer to have n, and c t transformed to mean elements. The
last term in equation (21) increases in magnitude, progressively with the time. The computa-
tion of this term of large magnitude may be avoided by modifications of the elements n, and c,.
The method of transformation can be clearly shown from the example (10) Hygiea,
T^wfl l(l**+^ > )(?*~^ n/te ^) = ( 6 - 80497 ) + (8.3192) [n'dz'] (31)
By equation (1)
(6.80497,)E + (8.3192) [n'dz'] = (6.80497 ,) c, - Of 4067 t - 14f6 sin E
+ (6.80497,)n<fcr + (8.3192)[/<Jz / ]
It is evident from equations (1), (21), and (23) that the first term on the right-hand side
of equation (32) may be combined with the mean anomaly at the epoch to form a mean mean
anomaly, given by Cj _ ^ + (6 .80497 Jc,
Furthermore, the second term on the right-hand side of equation (32) may be combined
with nt in equation (1). A mean mean motion is thereby introduced, which is given by
n, - n, - Of 4067 = 636f8566
Again, the third term on the right-hand side of equation (32) may be combined with a
term in (ndz [ndz]) which has the argument E. In the construction of (ndz [ndz]) there
occurred the terms + 34fg gin +4 , 6 co = (1 545) ^ ( + 7 o 53)
The addition of 14f6 sin E from equation (32) gives
+ 20f2 sin E+4f6 cos E = (1.320) sin (e+12?74)
These two values for the argument x = E are tabulated in the body of the table given on p. 27.
Further, since it is intended to improve the perturbations by the use of Xewcomb's value
for the mass of Jupiter, ndz must be multiplied by the factor 1.00050. The combination of the
correction for the mass of Jupiter with the term of the same form in equation (32) gives
( + 0.00050 -0.00064)7nJ2 = -0.00014 ndz
This correction is the last step in the determination of ndz, since it depends upon the pertur-
bation itself.
Without change of notation for ndz, the collected results are:
E e sin E = c 2 + ndz + nj (33)
where
ndz = [ndz\ + (ndz - [ndz]) - 0.00014 ndz + (8.319) [n'dz'] (34)
It must be remembered that [ndz] t and (ndz [ndz]) are numerically different from their
original values, but there should be no confusion if this transformation is not made before the
constant c has been determined.
The constant elements are now:
Epoch and Osculation, 1851, Sept. 17.0, Ber. M. T.
n, = 636f 8566 = 0? 17690461
c, = 121?8661
<P l = 6.3858
*, = 230.7971
ft = 2S7.6198
' = 3.7857
20 MEMOIRS NATIONAL ACADEMY OF SCIENCES.
Equinox and ecliptic, 1850.0
logP = 9.98741 log ^ = 9.04620 log p 2 = 0.49191
X= 9.95150
V-i _
fX7= 9.
log a* = 1.491 93 log a t = 0.49731
Certain other transformations of the elements which v. Zeipel makes are omitted. Those
terms of the perturbations which have the argument s have the same period as the planet and
can, therefore, be absorbed in the elements. It would be necessary to set up formulae for this
transformation to mean elements, and it is not profitable to do so.
PERTURBATIONS OF THE RADIUS VECTOR.
The perturbations in the radius vector are computed in a manner similar to that for
(nds [nUz]). In Table XLIII the numerical coefficients are multiplied by their respective
factors w*, -if, 9'', J*, the terms are collected, the known parts of the arguments are incorporated
in the coefficients, and the terms are grouped in the form:
v i vl
v = 2 a sin x + 2o cos \+
+ (tf-#o) ( 2a> sinx+JZ>' COSX+- } (35)
-K#-tM J Ua" sin x + 2-fc" cos x+ !*.!;!.} +
Let
a = It sin K I =Jc cos K
a' = Jc' cos K' b' =' sin K' (36)
a"= -V sin K" &"=-*" cos K"
Then
v = 2Jccoa (x+ft + W-flJZk'smb+IO + W-WWcos ( X +K") + - (37)
and to correct the perturbation for the use of the improved value of the mass, i> should be
multiplied by 1.00050.
If the mean motion n t is adopted, the constant in v must be corrected by
t 1
3 n t sin 1"
This correction of the constant in v permits the use of the relation
7i 3 2 a 2 3 = P
in the computation of a geocentric place; without this correction it would be necessary to use
the relation
nfaf = P
in the determination of the parameter p. In the computation of the eccentric anomaly it
is permissible to use either n l or n 2 , for the difference is taken up in the modification of 77^2,
but the theory of Hansen demands the use of constant elements. Hence, strictly speaking,
7i, must be used in computing a geocentric place. The modification of the constant in v renders
the employment of n 2 equivalent to the use of n t .
(10) Hygiea.
2 n,-ro, 1 _2 Of 4067 1 _,
3 n, sin l"~ 3 637f3 sin 1""
The constant in Table XLIII or C'is +47?6. Therefore, the new constant is:
+47?6-87f8=-40?2 = (1.604) cos 180?00
where the coefficient is logarithmic in seconds of arc.
The perturbation is tabulated on page 27.
NO. a.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 21
PERTURBATIONS OF THE THIRD COORDINATE.
The perturbations of the third coordinate are derived from Tables LTV, LVi, LVn or D,
E,, E,. The first of these is of the same form as the tables for (nSz [nSz]) and v. After mak-
ing analogous transformations and multiplying by the factor i cos i, (i is defined by equation (6)),
i cos il U p . q 7jPi ) '''siiiA = 2Jcsw(x+K) (39)
Both Table LVi or E, and Table LVn or E a lead to a single numerical quantity, since all
the factors and arguments are known constants.
The perturbation u is given by
= i cos i [2U p . q r)i>T)'i sin A + njt.{K l (cos e eJ + K, sin e} +c t (cos e e,) + c, sin e] (40)
in which c,, Cj, the constants of integration, have not been determined.
The constants c t and c, are determined by Hansen's conditions:
(41)
__ . _ III
dt
Substituting these relations and equation (39) in equation (40), the determination of c, and
c, is given by the solution of
C l (cosf-e 1 ) + C t sias=-IJcsin(x+K); C t sin e - <7, cos e = 2lc ^ cos (x + K) (42)
where <7, =- 1 cos i.c, , _.
C. = i cos i.e.
and
dx
d e
where
dfi l+<r w -
dt l+HA'+B, 1 )' 2
A double notation is used here, for cos i is the cosine of the inclination of the orbit, and is
M
the numerical coefficient of e in the argument x, but this should cause no confusion.
Dividing and multiplying the factor
i cos i-nj,
by 365.25
i cos i-n. rr>
1 COS V1 *- -365^5 T (45 >
where T\s the interval in Julian years, measured from the date of osculation.
It is evident that
C l (cos e ,) + C t sin e
can be incorporated in
Jit sin
in the same manner as similar terms were treated in (ndz [ndz\).
For symmetry of form, let
c cos i- nj{ KI (cos e-eJ + K, sin e} =2V cos (x + K') (46>
finally, then, without change of notation,
M = Jisin (x+ K) + TZk' cos (x+Jf) (47)
in which the constants of integration are absorbed in the first term. The perturbation u is
tabulated on page 27.
The perturbations in the heliocentric coordinates are computed from equations (3) The
signs of cos a, cos b, cos c are determined as follows:
cosa>OifO<8< 180
cos 6<0if -90 <& <+90
cos 5 < in any case, if e > i
cos c > if sin i cos ft < cos t
cos c>0 in any case if i<45
22 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.
(10) Hygiea.
t =
[(4.41940) cos (2Co+ 72?5246)
+ (3.9150) cos (4
+ (3.220) cos (6C + 186?48)]sin 1
S=- 285
Jfcsin (x+K)=- 70f5
<7
cos (x+K)= + 101f6
Ci- + 35?9
<7,= + 12058
From Table LIV. multiplied by t cos i we have three terms in
!-i iioiJoIoa ;flJ.vtf a<v/.^ * ^
Jfcsin (x+-K) = -4.'2-lf9sin + 2f7 cos e
which, added to
C",(cos e-ej + C 2 sin s= + 12058 sin + 3559 (cos e-c,)
gives for two terms in 2" sin (x+ K)
hfi/i
-758 + 11859 sin + 3856 cos =(0.89) sin 270?0+ (2.0970) sin (s + 17?99)
CHECK COMPUTATION.
After the elements have been determined and the final tabulation of the perturbations is
ready, the following checks should be performed, even if the computation has been duplicated.
t =
6 =|(e e sin s)q'
1 w
g' = c' + [n'dz'} 6 = t> + g (ndz - [ndz]) - yw sin
?.i.<Vi'. "1
where the necessary quantities are to be taken from the last approximation for c.
Secondly, the heliocentric coordinates
x-Ax, y-Ay, z-Az
for t = must check when computed by the usual formulae for two body motion and osculating
elements, and when computed with the final set of elements and the corresponding perturba-
tions, ndz and v, taken from the final tabulation.
The final tabulation of the perturbation in the third coordinate is checked by the test
t=Q u =0
(8*)
COMPUTATION OF THE PERTURBATIONS FOR THE TIME t.
It is well to emphasize here the distinction between the elements n, and c, and the elements
n t and c 2 in their relation to the perturbations. Let ndz l denote the perturbation in the mean
anomaly computed according to equations (20), (19), (18), (21), (22), (27), (28), and let nz 2
signify the perturbation computed according to equations (20), (19), (18), (22), the final tabu-
lation of (ndz [ndz]), and an equation analogous to (34). (It must be remembered that equa-
tion (34) is for (10) Hygiea only. The numerical coefficients are determined for each planet
individually.)
Before the determination of c there can be no confusion, for there is but one way to com-
pute the perturbation ndz. Later, when both c, and c 2 are given, the computation may be per-
formed in either manner. The latter method is, of course, adopted. The question then arises,
what values of and c are to be used in equation (19) ?
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
23
Clearly, there is only one value of e, for
s i sn
and both rufe and e must be found by trials. Further, since the introduction of n, and c, arises
merely from a transfer of certain terms in the perturbation, the argument of the perturbation
is independent of this transformation. Therefore c t is the constant in equation (19).
For any time t the order of computation is: equation (33), neglecting n9z, (20), (19),
(18), (22), final tabulation of ndz [ndz], and the equation analogous to (34). Since the per-
turbations are large, the argument is not sufficiently accurate when ndz is neglected. It is,
therefore, always necessary to make a second approximation for ndz. In the first trial the small
terms may be omitted.
(10) Hygiea.
PERTURBATIONS n8z, v, u, FOR 1873, SEPT. 20.4491, BER. M. T.
log , (degrees)
logf
0.80432
8. 48578
log sin
log sin
log sin
9.9387,
9.9918,
9.703,
1 2
97*;fil
Iog 57.30 w
. / tWIi
'2177278
2
Co
2207848
log cos
9. 741,
. *
Vjj
2217880
log cos
9.419
2
.
12178661
W (r r )
1.6337
;*]
2*
4*+ &
3157360
o p*>e ' *u
iog^:-: >
1.3898
Jf-l~3$
119. 524
is -^5^
283. 698
4- ; fli^t *; j <
1873
log ^(C-Co) cos
1.131,
-+ *
208.824
u
lr_l_3|}
12.998
2
Ber. M. T.
Sept. 1 20. 4491
log-(C-C.)coe
0.809
106. 526
(
+ 8039? 4491
270. 700
n^l
+ 142272156
11405"
E+40
74. 874
Cj+Hjf
nz
1544. 0817
104. 0817+1440
'3.666
"A + i A
2017
140
124
+6*
+ 8t>
239.048
43. 222
57 648
Jf=c,+n,<+na*
100. 416
+10
- f+4tf
221.822
*
i ' 106. 526+1440
\ 1546. 526
[*],
13676"
|+3a
61. 876
226.050
30 224
log*
3. 18935
log [n9z\ (sees)
log [nte], (degrees)
4.13596,
0. 57966,
|+7<>
194. 398
log^
1. 67513
[jufc],
-3. 7989
-Jt+ t
102.298
w
r
+ 477329
2c
213. 052
17.226
-['*']
+ 477053
log (9.6715) [jute],
0. 2512,
2+4.
181. 400
345.574
log fe-KAq)
1. 67259
(9.6715) [ndi],
-17783
|+W
136. 750
* . ' X
4*+7*
300.924
log (9.99572) (J^e- [n'dz'U
1.66831
a
2627087
(9.99572)(|e-[n'S^)
+ 467592
d-0
407207
r
2637870
f
2627087
2'
1677740
2$
164. 174
\"
335. 480
3i>
66.261
6;
143. 220
4t>
328. 348
5t>
230. 435
C~Co
43.022
M
132. 522
7j>
34.609
2+ 72?5246
2407265
M
2%. 696
4^+305. 627
281. 107
()
6^+186. 48
329. 70
i*
53. 263
!
106. 526
2;+ 68783
236. 57
1*
159. 789
4^+309. 75
285.23
2e
213. 052
i
266. 315
1 Con. lot aberr.
1 From previous approi.
* From Astrand's table.
See eq. (1), page 16.
24
MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.
(10) Hygiea.
PERTURBATIONS nSz, v, u, FOR 1873, SEPT. 20.4491, BER. M. T. Continued.
x-i-f+j<?
nSz [nit}
y
u
i i
x+K
log sin
(.x+K)
It sin (x+K)
X+*"
log cos
(x+K)
* cos (x+K)
x+K
log sin
(x+K)
* sin (x+K)
O
o
180.00
0. 000 n
- 40"
270.00
0.000 n
- 8"
2
58.608
9. 7167
+375"
296. 33
9. 952 n
-12
4
98. 841
9. 1867 n
- 33
238
9. 928 B
6
o
146. 28
9. 920 n
- 52
1 1
353. 286
9. 0679 n
- 56"
173. 425
9. 9971
- 137
80.40
9.994
+ 11"
1 3
41. 102
9. 8178
+ 479"
221. 824
9. 8723 n
- 193
110. 78
9.971
+ 14
1 5
88.71
0.000
+ 265
268. 86
8. 2988 n
- 2
156. 04
9.609
+ 3
-1 1
233. 74
9. 907 n
- 85
192. 38
9. 9898 n
- 3
122. 28
9.927
+ 11
-1 3
106. 53
9.982
+ 41
111. 41
9. 562 n
- 13
172. 10
9.138
+ 1
2
119. 27
9.941
+ 18
300.02
9.699
+ 3
124. 52
9.916
+103
2 2
347. 748
9. 3268 n
- 761
167. 726
9. 9900 n
-1852
79.94
9.993
+ 59
2 4
35. 927
9. 7686
+ 437
215. 194
9. 9123 B
- 329
104. 99
9.985
+ 25
2 6
83.53
9.997
+ 243
263. 148
9. 0767 n
- 15
150. 50
9.692
+ 5
2 8
127.4
9.90
+ 35
309. 92
9.807
+ 27
-2 2
115. 61
9.955
+ 84
88.6
8.39
0.51
7.948
+ 1
-2 4
311. 82
9. 872 n
- 3
349. 03
9.992
+ 6
3 1
276. 65
9.064
+ 1
225. 83
9. 856 B
- 5
3 3
163. 51
9.453
+ 36
341. 94
9.978
+ 85
257. 49
9. 990 B
- 3
3 5
208. 94
9. 685 B
- 34
28.42
9.944
+ 34
278. 56
9. 995 B
- 1
3 7
253. 08
9. 981 n
- 13
54.02
9.769
+ 1
-3 1
136. 98
9. 864n
- 2
4.98
8.939
4
348.8
9. 288 n
4 2
274. 434
9. 9987 B
- 104
40.38
9.811
+ 1
4 4
327. 82
9. 726 n
- 21
156. 79
9. 963 n
- 9
85.1
9. 998
+ 1
4 6
22.1
9.58
+ 5
198.96
9. 976 B
- 4
92.97
9.999
+ 1
5 5
150.8
9.69
+ 5
330. 79
9.941
+ 10
5 7
196.6
9.46 B
- 2
16.85
9.981
+ 8
+1648" -1079"
+550" -2684"
+236" -29"
(i O'orT
x+K'
log cos
(x+ff'J
k' cos (x+K')
x+K'
log sin
(x+K')
*' sin U+JT)
Cx+JP)
log cos
(x+K')
)c'cos(x+A")
o
270.00
0.000 n
- 6"
180.00
0.000 B
2
232.94
9. 902 n
g
4
o
281. 51
9. 991 B
- 7
2
292.53
9.583
+ 371"
292. 573
9. 965 B
- 447
47.67
9.828
+ 6"
2 2
4.7
0.00
+ 2
351. 93
9. 147 B
2 4
41.3
9.88
+ 6
41.29
9.819
+ 3"
-2 2
123. 85
9. 746 B
- 2"
305. 02
9. 913 B
- 1
4
219.90
9. 885 B
- 20
4 2
104.1
9.39 B
- 1
104.23
9.986
+ 4
4 4
154.82
9. 957 n
- 1
147
9.736
+ 1
+ 379" - 24"
+ 8" - 469"
+ 6"
(*-.)'
X+K"
log sin
(X+1T")
Jc" sin (x+K")
x+K"
log cos
<x+*")
V eta (x+K"-)
O
O
2
296. 23
9.953,
- 3"
112. 79
9. 588 B
- 1"
4
227. 15
9. 865 B
- 1
47
9.834
4"
1"
1 For perturbation u use factor T.
No. 8.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
(10) Hygiea.
PERTURBATIONS niz, v, u, FOE 1873, SEPT. 20.4491, BER. M. T. Continued.
25
log(0-!> )rad. 9.8462
/ .
log (> ->)* 9.692
log T 1. 3426
f.tj
log a sia 1"
5.183
cos t
1
n!z
r
u
2kaa(x.+K) + 569" Ik coe (x+^)
- 2134"
It sin (X+.BT) i + 207"
Jf cos (x+^O + 355
IV sin (x+A"0
- 461" ! JF cos (x+-K"0 6
IV sin (x+^'O 4
IV coe (x-f ")
1"
log IV coe (x+^O
log (^-^ ) If coe(x+ / K v )
2.5502
2.396
+ 249"
log IV sin (x+^0
log (d - >.) IV sin (x+^0
2. 664, log J'f coe (x+^"0
2. 510, log T. IV coe (x+-K"0
- 324" T. JF cos (x+^0
0.778
2.121
+ 132"
logJt"sin(x+^' / )
0.602,
r
- 2458"
u
+ 339"
0. 294,
log * (aecs)
3.3906,
logu
2.530
(t !>,,)*.
2"
log ^ (r^)
8.0762,
7.713
log (l+)
9.99480
log cos a
8.798,
log cos b
9. 619,
I
+ 816"
log cos c
9.958
1 Z J 1
+0. 2267
(8.3192) [n'Jz']
+0.0058
log Ax
6. 511,
[ri^z].
-3. 7989
log ^ J/
7.332,
tuJz
- 3.5664
logJz
7.671
-0.00014 TKte
+ 5
n*z
- 3. 566
Ax
-0.00032
if . ** ^'
Ay
-0. 00215
At
+0.00469
The computation of the geocentric place on page 26 is analogous to the usual method for
two body motion, the fundamental equations being (1), (2), (3). A complete set of formulae
and an example of the computation is also given in Memoirs of the National Academy of
Sciences, Vol. X, Seventh Memoir, p. 215.
CONSTANTS FOR THE EQUATOR.
A' yearly vr.
B' yearly var.
C' yearly vw.
log sin a log cos a
log sin b log cos b
log sin clog cos c
issao
1900.0
1950.0
3209833+0901399
321. 532+0. 01399
322.232+0.01399
22991g2+0901404
229.885-0.01405
230. 587+0. 01406
238657+0?01310
239. 312+0. 01308
239. 965+0. 0130S
9199914 8.799,
9.99914 8.797.
9.98915 8.795,
9.95884^9.619,
9.95868 9.619.
9.95853 9.620.
9.62355 9.958
9.62423 9.958
9.62490 9.958
26
uA hod.i'i
ifj VltlHiN
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
(10) Hygiea.
COMPARISON, OBSERVATION COMPUTATION. 1873, SEPT. 20.4491, BER. M. T.
[Vol. XIV.
1873
X
+3. 0709
Ber. M. T.
Sept. 20.4491
X
-1.00281
*a~f~ n j'
104?0817
dx
-0. 00032
n8z
- 3. 5660
*
+2. 0678
M=c^+n, i t+n8z
100. 5157
y
-0. 89314
dM
- 04843
Y
+0. 03260
dM'
- 29/06
jj,
-0. 00215
d<p'
+ 3/15
-0. 86269
dv
dip
+ 1. 8124
f X/flf^ j tff
+ 68
\ fl /
1 rl
z
-0. 19677
^V-SS
+ 8
Z
+0. 01415
d(v-M)ldM
+ 5/73
- 0. 0674
Jz
f
+0. 00469
-0. 17793
l hD m .dM
+ 8
)2.dM'
+ 1/94
log p cos S cos a
0. 31551
v-M
+ 12 0/14
cos a
9. 96515
M
f+ 12 7/81
sin a
9. 58550 n
Vi~ MI
1+ 12. 1302
log p cos 8 gin a
9. 93586 n
I/-*,
112. 6459
log tg a
9. 62035 n
J337 21' 14"
log cos/
9. 5S550 n
a
\ 22 h 29 m 24'. 9
log ! cos/
8. 63170 n
Red to True a
+1.5
log (1+e, cos/)
9. 98099
True a
22 h 29 m 26 s . 4
logr
0. 51092
Obs. a (A. N. 2029)
22" 29 m 07'. 1
log (1+K)
9. 99480
logr
0. 50572
log p cos d
0. 35036
"R'
C"
321?1548
229. 5058
238. 9584
cos 8
sin 8
log p sin 8
9. 99864
8. 89852 n
9. 25025 n
>')i. ;t'>v
log tg 8
8. 89989 r
7
73. 8007
8
-4 32' 26"
'+/
342. 1517
Red to True 3
+6"
C"+/
351. 6043
True 8
-4 32' 20"
Obs. 8 (A. N. 2029)
-4 33' 27"
log sin a
9. 99914
log sin (A'+J)
9. 98240
-)
.!:/ 'l;i'J<-t f.
logz
0. 48726
logp
0. 35172
log sin 6
9. 95877
log sin (*'+/>
9. 48643 n
logy
9. 95092 n
(0-C)
Act cos 8
-19-3
log sin c
9. 62387
J8
_!' T'
logsm(C"+/)
9. 16438 n
log 2
9. 29397 n
Stlit
Given a series of observations well distributed around the orbit and extending over as long
an interval as is available, the elements can be corrected by the method of least squares.
For this purpose the formulae by Bauschinger 2 are convenient. The equations of condi-
tion are set up for the residuals in the plane of the orbit and perpendicular to the plane, as seen
from the earth. This resolution of the residuals is convenient because it keeps the same reso-
lution into components as is used in the theory of Hansen.
It is to be noticed that the elements to be used in computing the differential coefficients
are the finally adopted constant elements referred to the equator by the proper transformation.
The value of r to be used is
except in the equation
sm
sin/
(Hansen's notation)
' Tafel zur Berechnung der wahren Anomalie, Veroftentlichungen des Rechen-Instituts der Koniglichen Stemwarte zu Berlin No. 1.
1 tiber das Problem der Bahnverbesserung, Veroflentlichungen des Koniglichen Astronomischen Rechen-Instituts zu Berlin, No. 23, Berlin,
1903.
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
27
The use of ,/, r and constant elements is equivalent to the use of osculating elements for the
given date of observation.
(10) Hygiea
^
^
V
u
( i
log It
K
logi
1C
log*
K
9
1.604
180.00
0.89
270.00
2
2.8570
254. 434
1.118
132. 16
4
2. 3364
130. 493
8.25
270
6
1.800
13.76
ndz [n8z]=2k sin (x+K)
1 1
2. 6771
37. 936
2. 1397
218. 075
1.057
125. 05
-|-( ) j_i} yjj - / k / cos (x-^-K'}
1 3
2. 8627
281. 578
2. 4135
102. 300
1.161
351. 26
+(# d<,Yk" sin (\+K")
1 5
2. 4238
165.01
1.965
345. 16
0.930
232.34
-1 1
2.022
24.92
0.55
343.56
1.119
273. 46
-1 3
1.628
93.53
1.543
98.41
0.981
159. 10
2
f [1.545]'
\ 1.320
[7. 53]'
12.74
0.711
193. 49
2.097
17.99
v=Ik cos (x+JiO
+(tf-i> )Zf sin (x+-K"')
2 2
3.5546
77.048
3. 2776
257. 026
1.777
169. 24
-(-(> i> ) J ^t" coe (x+^'O
2 4
2. 8719
321. 053
2.6054
140.320
1.412
30.12
2 6
2.389
204.49
2. 1033
24.100
1.034
271. 45
2 8
1.64
84.2
1.62
266. 70
-2 2
1.970
57.96
1.27
31.0
1.824
302.86
u= Ik sin (x+JQ
-2 4
0.602
90.00
0.80
127. 21
+ T2V cos (x+-ST')
3 1
0.90
214. 77
0.826
163.95
3 3
2.100
297.46
1.95
115. 89
0.446
31.44
Where T is expressed in Julian years
from date of osculation.
3 5
1.841
178. 72
1.583
358.20
0.171
248.34
3 7
1.12
58.68
0.34
219. 62
-3 1
4
4 2
2. 0170
257. 208
0.42
34.68
0.673
0.00
0.270
262. 68
135.7
23.15
x=tW+y* where in s the multiples of
2r must be retained.
4 4
1.589
146. 42
0.97
335. 39
9.91
263.7
<> =221.811
4 6
1.14
36.5
0.66
213. 39
9.73
107. 40
5 5
1.038
14.0
1. 062
194.04
5 7
0.88
255.7
0.94
75.93
(-i,) or r log V
K' log it'
K'
log*'
K'
0. 799
270.00
9.690
180.00
'
2
1.021
68.77
4
0.86
313. 16
2
2. 9862
186.00
2.6850
186. 047
0.957
301.14
2 2
0.18
94
0.12
81.23
2 4
0.88
326. 4 0. 60
326. 42
-2 2
0.60
66. 20 0. 11
247. 37
4
1.414
6.85
4 2
0.68
86. 9 0. 580
87.00
4 4
0.11
333. 42 0. 09
326
" v .
(>-*,) log *"
K"
logi"
K"
.
2
0.58
189. 70
0.26
6.26
4
9.91
14.10
9.6
194
COMPARISON OF THE REVISED WITH V. ZEIPEL'S ORIGINAL TABLES.
It was originally planned to conclude the example with a least squares solution of the orbit
on the basis of the observations used by v. Zeipel for the same purpose, and to test conclusively
the relative value of the revised and v. Zeipel's original tables by representing recent observa-
tions with both sets of elements and tables.
In the course of the computation doubt arose regarding the accuracy of some of the
observations selected by v. Zeipel, which led us to reject them and substitute other observa-
1 In the determination of the constant e use quantities in brackets.
28
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
tions. This substitution produced an unfavorable distribution of the observed places in the
orbit and the resulting least squares solution was not satisfactory.
In the meantime, pending a resumption of the least squares solution on the basis of a more
favorable distribution of observed places, 1 the following conclusions may be drawn regarding
the revised and v. Zeipel's original tables:
1. v. Zeipel's tables have been slightly improved by the correction of some numerical errors.
2. A moderate further improvement has been accomplished by an extension of the tables
in so far as seemed practicable without a more exhaustive and unwarranted study of the prac-
tical convergence of the auxiliary series, by including certain terms of higher order and degree.
With reference to the correction of the orbit and the representation of observations by a
least squares solution, it should be observed that
(1) A symmetrical distribution of the observed positions in the orbit is essential to coun-
teract the effect of neglected perturbations of higher order and degree and of major planets
other than Jupiter. For the Hecuba Group, in general, the mean motions of the minor planets
may be nearly commensurable with those of Saturn, Mars, or the Earth in the ratios 3/2, 3/1,
or 3/5.
(2) However accurate the initial osculating elements may be, comparatively large residuals
may remain on account of neglected perturbations.
Logarithmic.
TABLE A (XXXV).
n&z [niz]
Unlt-1"
Sin
te-i
W-*
--
jf
w
to'
tf
,. +
4. 1570
4. 8741 B
Jf+ 0+ 4
2. 7684 B
3. 3827
3. 7172 B
B*
Je+ (j-j. j
4. 0056 n
4.7686
*"
J + t>+ 4
4. 0766 B
4. 8295
J-+ 0+ 4
4. 1365
4. 8738 B
' '<
+ 0+24
3. 3345
4. 5162 B
j: + 30+24
4. 2240 n
4. 9611
5. 6685 B
]J
Jtf+30+34
4. 0671
4. 8483 B
5. 5636
1J /3
i +50+34
5. 0926 n
6.0018
Jf
I '+50+44
5. 2325
6. 1714 n
i+50+54
4. 7675 n
5.7344
/"
j+50+44 2
3. 8050 n
4. 7998
5'
-$+
3. 3112
3. 8350 H
4. 1355
^r-f- t5-f- A
3. 2065 n
3. 7910
4. 0833 B
lj"
-}+30+ 4
3. 5338
4. 6236 B
ll'
j+30+24
4. 0879
5. 0382
1J
if+30+34
tv ' .
3. 6012 n
4. 5318 n
J 1
-if+30+24-J 1
3. 2074
4. 1925 B
,
,
9. 868 n
0. 5689
2.922
3. 4600 B
3. 3670
Jj'
+ 4
9.482
0. 2533,,
2.673^
3. 2959
3. 1772 B
5 1 ?'
+20+ 4
0. 746 n
1.384
3. 2927 B
4. 14906
4. 6990 B
+20+24
9. 788 B
2. 47560
3. 10847 n
3. 4540
3. 3960 B
n>
+20+24
0.645
1.342 B
2. 305 n
3. 6179 n
4. 4018
*"
t+20+24
0.326
1. 119 B
2. 935 n
3. 3017 n
4. 39206
+20+24
3. 4276 B
4. 23764
4. 76933n
'!')'
+20+34
0. 28 B
1.102
3. 1738
3. 5449 n
3. 8446 n
rjij' 3
+40+24
3.6004
4. 27485
1
(.(-40+34
9.057
0. 692 B
3. 10161
3. 9302 B
4. 52415
4. 78162 B
,V
+40+34
4. 0519 n
3. 7975
yf
+40+34
4. 1385 n
4. 6961
+40+34
4. 2431 B
5.1290
17
+40+44
9.500 B
0.522
2. 9351 B
3. 8035
4. 41616 n
4. 63017
if
+40+44
3. 7714
4. 2108 B
Iff'*
+40+44
4. 4165
5. 0931 B
Pi)
j-j-4,j-|-44
4. 1524
5. 0661 B
,y
+40+54
4. 0588 B
4. 8136
' Since 1913, when the revision of the tables was concluded, Miss Glancy has continued the problem of ( 10) Hygica independently at the Observa
irio National, Cdrdoba, with the following highly satisfactory results, which substantiate further the increased accuracy of the revised table:
va-
torio Nacional, Cdrdoba, with the following highly satisfactory results, which substantiate further the increased accuracy of the revised tables
(1) The original osculating elements and the revised tables resulted in a greatly improved representation of the selected observations (1849-188i)
over the representation obtained with the original tables. (2) After the correction of the original osculating elements by least squares solution
(a) on the oasis of v. Zeipel's tables and residuals, (6) on the basis of the residuals resulting from the revised tables, the representation ol the
selected observations was equally satisfactory; but 3 later observations, taken in 1910, 1914, and 1917, are represented far better by the revised
tables and corresponding elements than by the original tables and corresponding elements, (of. Astronomical Journal, Vol. 32, p. 27, No. 748,
January 1919) A. O. Leuschner.
Ko.3.1 MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
Logarithmic. TABLE A (XXXV) Continued.
29
Unit-l"
Sin - K"- 1 M
V 1C*
J 1J
f-J-4^-f-3J 2
3.2322,
4.2342
c-f~4iJ-|-4J E
2.744,
3. 0962
,/S
t+6t>+4J 0.28 0.64,
3.8027
4. 77998, 5. 52852
- -/
-j-6iJ-(-5J
0.596,
1.070
3. 9374,
4. 94342 5. 70347,
ij l
c -\-M+<}J
0.255
0.8,
3.4684
4.50125,
5.27451
+6t?+5J-J
8.8
9-3,
2.415
3.4823,
4.2931
V*
e+8^+5J
4.5564
5. 4999,
"7
-j_g^_j_6J
4.8668,
5.8416
_)-g^_(-7J
4.6990
5. 7030,
i*
e-(-8i>+8J
4.0631,
5.0844
j ^'
j-Lg^-LgJ^ 1
3.5829
4. 6352,
ft +8e>+7J-J
3.3768,
4.4540
l"
- +2<J
0.606
1.422,
3. 2132
3. 6657,
3.9260
r '
+ 2l?+ J
0. 791, 1. 690
3. 3777 B
3.8866
4. 72168
1
- +2tf+2J
0. 418 1. 365,
2.894
3. 4616..,
3.8078
- +20+ A-I
9.34
0.28 B
2.938
3. 4714,
3. 7862
/i
- +4^+ A
' t '->
' -* ,'
3.5208
4.07255
7 r/' 3
3.4965,
4.59582
r*r'
_ -j-4ty+3J
3.2416
4. 5467,
1)' - +4tf+4J
2.430,
3.9848
,' - e+4d+2J-2
3.5496
4.19852,
- f+4<5+3J-J
3. 3247,
4.05994
55'
^ +3l j + 2J
3. 6731
4.0029,
|+3tJ-(-3J
2.3528
3. 2475,
3.9005
ij'
^+3i>+3J
3. 6181,
4.2122
f
-j-3,y-j-3J
3. 4072,
4.4000
i + 3l?+4J
3.5244
4.4012,
7/ '
^+5t>+4J
3.3533
4. 4231,
5.2725
^
i+5t>+5J
3. 1780,
4.2730
5. 1359,
q'l
,-)-7i)4-5J
4.2775
5.4708,
jj -'
i + 7l? + 6J
4. 4051,
5. 6177
>z 2
i*-J-7iJ+7J
3.92%
5.1605,
ij
2^+2^+2^
9.486
2. 1744, 2. 708
2. 889, 2. 599,
' /
2-i-2i'+3J
1.946,
2. 501 2. 516,
? ?'
2 -f 4^-j-3J
8. 8, 0. 561
2. 789,
3.5813
4. 1074,
2j-j-4<>-[-4j
8.90,
9.599
1.711
2.5795,
3. 1726
*
2 +4^+4J
9.2
0.34,
2.618
3.4962,
4.0890
^'
2-j-6J+5J
9. 819,
0.5840
2. 7821
3. 7794,
4.51865
1
2 e +6<J+6J
9.653
0.4645,
2. 5979,
3.6265
4.38424,
Sr-j-otf + oJ
1.2340
2. 1166,
2.7076
T'
i-|-7i)4-6J
2. 3679
3.3518,
4.0587
1
|+7!> + 7J
2.1758,
3.1926
3.9204,
(t) #) coe
i;
0. 1021,
0.728
2.8978,
3.4504
3.7168,
1.377,
2.346
3. 8211,
4.6762
'i"
f
1.941,
2.815
4. 4076,
5. 1971
1.364
2.220,
4.4076 5.1971,
5. 7086
^'
e-f J
9.658
0. 774,
2. 7836 3. 3840,
3.6946
'/' T /
+ -1
1.863
2.755,
4. 2546 5. 0814,
*
*4" J
L844
2.642,
4.1953
4. 9770,
f *,'
+ J
1. 170 n
2.049
4. 3715,
5. 1770
5.6975,
v
+ 2J
1. 742 B
2.574
4.0203,
4.8466
j 3 V
t+ J 0. 716
1.65,
4. 0809 4. 8829,
5.4008
j*,
+ J+^
1.00,
1.89
4. 3427, 5. 0837
5.5553,
v
- J- J
1.562
2.455,
3. 9535 4. 7803,
,1
2:
9.801
0.43,
2. 5842 3. 1493,
3.4158
1 1
2+ J
9. 357 n
0.473
2.4548,
3.0830
3. 3936,
\v W 0' *-**
7 *
9.56,
0.42
+ J
9.43
0.32,
where C,,
sin Arg.4-( I )-iJ )J'r / P^ / 9; 2 Cj coe
2 , C 3 represent the respective coefficients.
sin Arg.
30
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
IVol. XIV.
Logarithmic.
TABLK B (XXXVIII).
*()
Unit 1 radian.
Cos
,,
JM
-.
M
'**
*-.
*
w
te*
1.5
3. 909 B
4.960
6. 6748 B
7. 2764
7. 540 B
7.31
**
2.0
4. 644 B
5.160
6. 150
8. 048 B
8.838
8. 655 n
8. 100 n
V*
1.9
3.41 n
4.75 n
6.509
8. 2077 B
8.994
8. 919 B
P
2.83 n
5.146
6. 299 B
7. 994
8. 740 n
8.656
*
2.34 B
4.446
4.57
6. 728 B
8. 4022
9. 1999 B
9.0854
8.079
Tj Tj
2i>
1.6
2.6 B
5.744
6. 535,,
8.3811
9. 1031 B
9. 0128
if
2<>+ A
0.8 n
3.068
5. 2988
7. 2212 B
7. 3772
8. 0372
8. 764 B
8.668
Tj 1)
2<?+ J
2.32 B
3.30
5. 886 B
6.718
8. 5059 B
9.2804
9. 201 7 B
1)'*
2t?+ 4
5. 301 B
6.149
8. 2302 B
9. 0154
8. 938 B
P y'
20 + A
8. 5592
9. 3245 B
9. 2428
y 3
20 +2J
2.48
3.40 n
5.422
6. 292 B
7.476
8. 664 n
8.636
Tj
20+2J
1.22
2.94 n
5. 1206 B
7.6416
7. 9638 n
7.083 B
8.645
8. 582 n
1) I)' 2
2#+24
1.9
3.0 n
5.442
6. 328 B
8. 0915 B
8. 630 B
8.742
3 V
20+2J
8. 5904
9. 3489
9. 8024 n
9.6532
TJ ij
2t>+3J
2.04 B
3.00
4.98 B
5.89
8. 0326 ! 8. 1973 n
7.69
fi)
2i>+ J-J
4.51
5.42 B
8. 1011
8. 873 B
8.792
.'2 _/
y v
20+2J-.T
4.04 B
5.00
6.89 B
8.182
8. 158 B
,'
40+24
2.66 n
2.7
6. 1031
8. 4188 B
8. 5297
6.0
7.90 n
Tj Tj
4<?+3J
2.72
4.369
6. 2526 B
8. 5594
8. 7988 B
7.94 B
8.287
8.210
!J 2
40+44
2.20 B
4. 624 B
5.824
8. 0924 B
8. 4333
7.24
7.74 B
8. 044 B
P
4<H-3J-.y
1.5 B
2.45
4.68
7. 1747 B
7.301
8.111
8. 127 B
6.J+34
5. 301 B
6.149
9. 1294 B
9. 7728
9. 6609 n
6i>4~4J
5.92
6.74 B
9. 4432
0. 14644 B
0. 05077
7)V
6i5+5J
2.0 B
3.0
5.93 n
6.79
9. 2774 B
0. 03298
9. 9494 n
n*
6<>+6J
2.0
3.0 B
5.420
6. 292 n
8.634
9. 4351 n
9. 3608
;? i 7
6<>-|-4J 2"
4.04 B
5.00
8. 272 B
9. 1028
9. 0334 n
J >!
e^+SJ-J
4.51
5.42 B
8. 0554
8. 926 B
8.864
(i>-<5 ) sin
^'
J
2.60 n
4.71
5.94 B
6.507 B
6.606
,/
2.+ 4
1.36
2.48
4.49
5. 255 n
5.51
5. 25 B
*
2t>+2J
1.82 B
2.42
4.64 B
5.350
5.51 B
5.16
,'2
40+24
2.34
3.00
5.392
6. 179 B
6. 528 n
6.665
Tj if
4i?-|-3^i
2.89 B
3.46
5. 702 B
6.467
6.851
6. 979 B
I*
4i?-j-4^
2.66
3. 459 B
5.357
6. 127 B
6. 530 B
6.653
1*
2.08 B
2.08
5. 546 B
5.546
if*
2.54
2.54 n
5. 396 B
5.396
!l'
4
2.5 n
2.5
5.776
5. 776 n
m' 3
m' 3
m' 3 , m' 2
m' 3 , m' 3
m' 2 , m'
m "' m/
m' 2 , m'
m /1 , m'
m' 1 , m
1 cos Arg.+( t ?-tf )Jit'*jP., / 9./.C 2 sin Arg.-f (^-t
where C,, C 2 , C 3 represent the respective coefficients.
cos Arg.
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE C (XLIII).
31
Ixxrarithmfc.
Unit-l"
Cos
-
2
-
w*
tr
8.72
9.88,
1.6349
2. 1070,
2.2333
f
9.80
0.212,
2.759
3.4922,
V*
8.9
9.23
2.937
3.6295,
'
J
9.66n
9.78
2.937,
3.1136,
3.6295
3.8440
If"
M
0.556,
1.204
3. 2111,
3.7970
If
2tf+ 4
0.504,
2.3472
2.456 n
2.686,
3.4735
rY
2i+ ^
0.997
1.711,
3.6559
4. 3103,
7"
2iJ+ 4
0.438
1.220,
3.3654
4. 0763,
i* *
2<>+ A
3.6975,
4. 3810
i)
20 +2 J
0.438
2.952,
3. 2529
3.0689,
3.3979,
f
2.5 +2 J
0.732,
1.497
3. 2410,
4.0643
7-j"
2tf+2J
0.772,
L589
3. 4136
4.0723
A
2J+2J
3.9048
4.5649,
4.9303
V
2i>+34
0.505
1.344,
3. 4757,
2.783
/*5
2+ J-2 1
9.33,
0.15
2.938,
3.5830
? V
2^+24 -J
9.20
0.10,
2.0251
3.2961,
?"
4J+24
8.9
1. 2819,
3.5514
3. 6173,
3. 8147
it*
w+w
9.75,
1.5024
3.7885,
4.1394
4. 3110,
r,>
4i>+4J
9.98
1.1342,
3.4007
3.9091,
4.1480
'V
4j+3^-J
6^+SJ
0.438
9.64,
1.220,
2.305
4.2675
2.542,
4.7993,
2. 749,
u?
W+4J
L125,
1.862
4. 6479,
5.2324
*v
6i)+5J
1.198
1.947,
4.5397
5.1768,
7*
6J-I-6J
0.732,
1.508
3.9457,
4.6328
F *'
6rf+44-^
9.20
0.10,
3.4099
4. 1710,
w
6tf+5J-J
9.70,
0.56
3.2601,
4.0542
9 v'
i+ i>
3.4878,
4.1106
i+ + J
8.3.
2.2106
2. 7179,
2.919
f
i*+ J+ 4
3. 5709,
4.2261
,
1 + tf+ J
3.4507
4.1296,
V
<H- rf+ ^
3.5100
4. 1837,
?V
v+ <J+2J
2. 579,
3.9270
7'
< t+3*+24
0.08
3.6873
4. 1471,
4.7839
q
. E+3J+3J
9.5
3. 5727,
4.1511
4.7545,
5"
< t+5tJ+3J
4.5568
5. 1414,
-M'
- t+5t>+4J
4. 7261,
5.4067
i 3
1 f+StJ+54
4.2862
5.0418,
J 3
1 t+5i+4J-J
3.2570
4.0005,
1
-i+ *
L086,
2.7090
3. 3467,
3.7098
1
-i+ l>+ J
0.88
2. 1967,
3.0952
3.5836,
I*
-i-t+3*+ J
2.514
4,1049,
ijr
- t+3J+2J
4.0853
3.9122
f
-i+3J+3J
3.8341,
3. 8118
f
-|+3J-f-2^-J
2.416
3.6926,
>!
i
9.62
0.58,
2.143,
2.682
2.9151,
r
e+ J
9.04,
9.9
2.061
2.666,
2.9477
if'
+2l>+ J
0.444
1.1661*
3.0588
3.8035,
4.2554
t+2^+2^
9.487
2.1744,
2.7280
2.972,
2.976
i 3
+2^^-24
0.344,
L1143
2.692,
3.5334
4.0772,
*"
+2J+2J
0.025,
0.828
2.634
3.0726
4.0416,
j 3
+2^+24
3.1265
3.8806,
4.3473
IT*
+2<>+3J
9.98
0.811,
2.873,
3. 1697
3.5856
t7
E+4J+24
1.105
L89,
2.864
4. 3477,
<
+4t>+34
8.8,
0.398
2.8000,
3.5327
4.0065,
4.3207
iV
+4t>+3J
1.260,
2.083
3.0931
4.4160
i"
+4t>+34
3.8375
4.0446,
>* i 7
+4tf+3J
0.267
1.15,
3.9421
4. 6972,
i
+4t?+4J
9.19
0.248,
2.6356
3. 4317,
3.9469
4.2558,
f
+4tf+4J
0.774
L66,
3.0934,
3.7866,
i"
+4J+4J
4.1154,
4.5547
j 3 '?
+4^+4^
0.455,
1.32
3.8518,
4.6436
v
+4^+54
3. 7579
4.3244,
/N
+4tf+3J-J
3.0030
3.8869,
32
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE C (XLIII) Continued.
Logarithmic.
[Vol. XIV.
Unlt-1"
Cos
w->
w->
w-i
to'
w
w'
f if
s+40+44-1
2. 4425
1. 85 n
^
+C^+4J
9.98 n
0.480
3. 5016,,
4. 3723
4. 9952 n
w'
e+6<)+5J
0.296
0. 823 B
3. 6369
4. 5582 n
5. 2093
f
+G^+6J
9.95 n
0.538
3. 1685 n
4. 1334
4. 8131 n
f
f+Gtf+SJ-I 1
8.5 n
9.15
2. 114 n
3. 0881
3. 7886 n
^
+8tf+5J
4. 2554 n
4. 9349
,,"
+8!?+6J
1.320
2. 152 n
4. 5657
5. 3010 n
,V
e+8,?+7J
1. 228 B
2.093
4. 3995 n
5. 1827
1*
e+8,9+84
0.648
1.54 n
3.7543
4. 5812 n
P l'
+80+64-.?
3. 2818 n
4. 1442
A
e+Stf+74-2 1
3. 0763
3. 9759 n
*"
- +2t5
0.305
1. 1007
2.912
3. 4958 n
3. 8151
11'
- +2<J+ 4
0. 490 n
1. 3330
3. 0166 n
3. 7273
4. 3119
V*
- e+2t?+24
0.117
0. 982 n
2. 288 n
3. 2375 B
3. 7892
f
- +20+ J-2
9.04
9.96 n
2.636
3. 2817 n
3. 6568
,"
- +40+ J
3. 2197
3. 9650 n
,,"
- +40+24
1. 146 n
1.89
3. 0204
4. 2441
*V
- +40+34
1.005
1.78 n
3. 5247 n
4. 0012 n
r f \i
1*
- +40+44
0.290 n
1.15
3. 1793
2.982
n- ^
f rf
- +40+24 -.T
3. 2486
4. 0585 n
:.-W-
S,
- +40+34-1
9.98 n
0.8
2. 957 n
3. 8580
y
|+ 0+ 4
9.0
2. 3363
3. 0704 B
3. 5111
r,'
i*+ 0+24
9.5
1.500
2. 3585
3. 1842 B
1)1)'
-|+30+24
2.779
3. 7820 n
^+30+34
9.28
2. 1614^
3. 0257
3. 6491 B
?
| +30+34
1.32
2.966
l"
|+30+34
3.3450
4. llll n
?
|+30+34
3. 2309
4. 1965 n
?Y
|+30+44
,
3. 2994 n
4. 1520
T,'
|+50+44
1.017
3. 1617 n
4. 1967
5. 01GO n
T)
|+50+54
0.88 n
2. 9688
4. 0380 n
4. 8781
*"
^+70+54
4. 0855 B
5. 2422
>y
|+70+64
4. 1991
5. 3823 n
^
fs+70+74
3. 7114 n
4. 9188
; 3
|+7d+64-J
2. 615 n
3. 8317
riV
-!e+ tf
3. 2411
3. 7872 n
tf
-i+ 0+4
2. 819 n
3. 4476
?
-^+ 0- 1
2. 9181
3. 4813
>?
2
B 98H.(
2. 364 n
3. 0737
v
2s+ 4
XX .0
2.624
3. 3489 B
,"
2s+ 24
2. 207 n
2.978
f
2+ 4+1
2. 620 n
3. 2765
1?
2s+20+24
9.8 n
1.63
2. 362 n
2. 873
*'
2 +20+34
9.5
1.796
2. 303 n
2. 1007
-M'
2 +40+34
1.93.
2.700
2 +40+44
8.7
8.8
1. 5802 n
2. 4158
2. 9867 B
*
2+40+44
2.330
3. 1764 n
,"
2t+40+44
3. 1079
3. 9008 n
;'
2e+40+44
2.736
3. 6809 n
V
2 4 +40+54
2. 9881 n
3.8425
V
2+60+54
9.64
0.53
2. 652 n
3.6204
4. 3279 B
>j
2+60+64
9.48 n
0.36 n
2. 4419
3. 4512 B
4. 1892
>)"
2+80+64
3. 6135 n
4. 6784
-M'
2 +80+74
3. 7124
4. 8075 n
5 1
2t+80+84
3. 2109 n
4. 3338
j*
2 +80+74-2 >
2. 068 n
3. 2092
i+50+54
9.3 fl
1. 140
2.0056
2. 5727*
*
^+70+64
0.5 n
2. 2749 n
3. 2377
3. 9184 n
>)
|+70+74
0.3
2. 0542
3. 0565 n
3. 7710
^+70+74
8.1
0. 4:) n
1.346
1. 959 B
(0-0 ) sin
.
nf
4
9.66
0. 810 B
2. 7559
3. 3840 n
3. 6946
r/
20+ 4
9.79
0.54
r;
20+2J
9.92
0.63 n
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE C (XLIII) Continued.
Logarithmic.
33
UnJt-1"
(>-t?i) sin
^
to-* _
w*
W
r
9.801,
0.425
2. 5970,
3. 1493
3.4158,
1*
t
1. 075,
2.045
3.5201,
4. 3751
11"
t
1.640,
2.514
4.1066,
4.8961
r>i
t
1.063
L916,
4.1066
4.8961,
5.4076
*
+ 4
9.36
0.471,
2. 4824
3.0830,
3.3936
Tj*1)
*+ J
1.565
2.456,
3.9671
4.7890,
Ij"
e-f- J
1.543
2.341,
3.8942
4.6760,
f ^
+ 4
f+ 2J
0.87,
1.441,
1.75
2.273
4.0705,
3.7192,
4. 8759
4.5456
5.3965,
f *l'
+
0.42
L36,
3.7799
4. 5819,
5.0998
I* 7 !
*+ J+Z
0.695,
1.585
4. 0417,
4.7827
5.2543,
|
t+4i>+4J
9.59,
0.45
f
t+4.+34
9.46
0.34,
1
2 f +2i+2J
9.45
0.11,
*'
2t+2,>+34
9.32,
0.04
Y
- t+ 4
L255,
2.149
3.6240,
4.4615
(-)<
i
,
9.25
0. 117,
''
+ -i
9.12,
0.02
m"
m"
m", m'
m'
m'
m'
j
COB Arg.+(*- l )JwijlV; 5 C', sin
where C,, C 2 , C, represent the respective coefficients.
110379 22 - 3
C, coe Aig.
Ml
34
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
:- TABLJB D (LIV).
2 U p . q r)Pii'V sin Arg
[Vol. XIV.
Logarithmic.
Unlt-1".
Sin
to-1
w*
U)
7)
- 4-n'
3.06^
3. 7258
v
-n'
2! 8235
3. 5528 B
/
20+ 4-n'
2. 2831
2. 8483 n
n
40+34 -H'
1.705
3. 1591 B
3. 9166 n
V
40+24-n'
3. 2462
3. 8608
n
j+ -n'
3. 2112 B
3.8544
v
j:+ 0_|_ 4 n'
2. 5875
3. 4153 B
/
j+30+24-n'
2. 2787
2. 6304 B
7;'
if+50+34-H'
3.3J55
3. 5865 B
if +50+44 -n'
3. 0779 n
3. 3972
li,
-i- 0-24-n'
-if- 0- 4-n'
3. 1158 B
3. 1493
3. 7378
3. 7644 B
,
-j + -n'
-j +30+ 4-n^
2. 3242
3.3863
3. 0060 B
4. 1833 B
T;
3. 3532 n
4. 1452
7)
+20+ 4-n'
2.6364
3. 3704B
3.8423
v
t+20+24-H'
1.423 B
2.706
3. 4014 B
+40+34 n'
1.4042 n
2. 1720
2. 6339 n
^/
+60+44 -H'
2. 3306 B
3. 1922
3. 7582 n
7)
+60+54 -H''
2. 1137
3. 0138 B
3. 6101
- -20-34 -H'
2. 7175
3. 4858 B
3.9484
T/
- -20-24-n'
2. 7756 n
3.5070
3. 9456 B
/
c 4 n'
1. 6810
2. 2463 B
_/
- +20 -n'
2. 8125
3. 4427 B
3. 7846
1)
- +20+ 4-n'
2. 9121 B
3. 4958
3. 8338
7)
$+30+24 -H'
2.6058
3. 5312 B
v
4+30+34 n'
1.760
1.82 B
*+50+44 n'
1. 7510 B
2. 8113
v
$+70+54 -n'
2. 9120 B
4.0813
_s _30_44_n'
2. 8673
3. 8458 n
v
IE 30 34 n'
2. 9620 B
3. 9124
-$- 0-24-n'
2. 0569 B
2. 7932
7)'
if+ 4 n'
2. 9275 B
3. 4708
-je+ -n'
2. 9702
3. 5487 B
1}
2 +40+34-n'
1.640
2.7S1 B
2j+40+44 n'
1.617
2. 340 n
2+60+54-H'
1.206 B
2. 2110
7]
-2 -40-54 -n'
2.4012
3. 3634 B
V
-2j-40-44-n'
2. 5241 B
3.4544
-2-20-34-n'
1. 5290 n
2. 3210
7)'
-2e -24 -H'
2. 3174,,
3. 0558
-2 - 4-n'
2. 3514
3. 0737 B
tTl'
u
I COS t
=2 U P . q i)Pij'<l sin Arg.+jt JiT, (cos t- e i)+K a sin f
- i)+Cj rin t.
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE E, (LV,).
Logarithmic. K l UnJt-1'.
COB
P
V
1*
1
to
UUU.UUU
+++ ++ 1
ntaaaaa
2. 9180 B
1.9821
2.8036
3. 5175
3.1764*
3.4580 n
3.7732
2.5473,,
3.n82n
4. 3017,
3.9772
4. 2668
2.8138
m'
TABLE
t COB Arg.
(LV n ).
Logarithmic.
t COS \
Unit-1".
Sin
...
u>
*
y'
4+n'
2.9180
3.7799
1.9821*
3.7744*
3. 5175*
3.4580
3.1764
3.7732,
4. 5819*
2.5473
4.5420
4.3017
4.2668*
3.9772,
2.8138*
m'
sin Arg.
'9 ain Aig.+niT,(cos t e)+K t ein |+e,(coe e)+Cj ain
35
i sw i-
! n O!f*.b
n'"
36
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
Logarithmic.
TABLE F (LVI)
w w a
Unit 1 radian.
Cos
Ml
->
UJ-l
u-o
W
at
4.360
5. 1966 n
5. 7767
r
4.766
6. 6599
7. 3732 B
7. 7492
2r
4.446
7. 1194
7. 7572 B
8. 0553
sr
4.412
6.8442
7. 5458 B
7.9060
4F
4.484
6. 588,3
7. 3450 n
7.7602
5r
6. 3437
7. 1490 n
7.6136
7T
5.875
6. 7632 n
7.3134
9o
-5r+20 +2J
6.5090
6. 6325 n
7. 4746 B
-4r+20 +2J
-3r+20 +2J
4. Win
3.19
6.169
6. 882]
7. 0658
7. 6078
7. 86980,
7. 9975 B
-2r+20 +2J
3.52
7. 098fi n
7. 6970
7. 9394 n
- r+20 +2J
5. 1420
6.359
7. 0722 n
7.4480
20 +2J
4.379
7. 6355 B
8. 2144
8. 4125 B
r +20 +2J
4. 856 n
8. 0894 B
8.9548
9. 5668 B
2r+20 +2J
C A
4.92
7. 8150 n
8. 6561
9. 2006 B
3r+20 +2J
5. 5174,
7. 6056 n
8.4650
9. 0111 B
4r+20 +2J
5. 4248 n
7.4128 B
8. 2958
8. 8561 B
5r+20 +2J
7.2254,
8. 1426
8. 7346 B
7r+20 +2J
6. 8746 B
7. 8484
8. 4936 B
J
-5r+20 + J
I^rf i
6. 8776,,
7.5604
7. 8425 B
4f +20 + J
! _tvjfl? .1
4.582
6. 8815 n
7.4536
7. 5238^
-3r+20 + J
M'*T .
4.674
6. 6271 B
6. 7816
7. 3174
-2r+20 + J
n<3ri<5 .il
4.99
6. 7985
7. 4732 n
7.7966
- r+20 + J
5. 4623 B
20 + Jo
4.605 B
7. 1987
7. 8314 B
8. 1061
r+20 + J
2r+20 + J
5.0056
4.38
8. 2964
8. 0434
9. 1086 B
8. 831 6 B
9.6833
9. 3296
sr+20 + J
5. 6251
7. 8458
8. 6564 n
9.1558
4r+20 + J
5.5812
7.6603
8. 5030 n
9. 0248
5r+20 + J
1 IH t-it>' v .' 1*
7. 4778
8. 3544 n
8.9050
7r+20 + J
" "* t S v (
7. 1130
8. 0545 B
8. 6668
I*
n t -.f( -1>T>,
4.664
4.71
5.83
r
7. 8102
8. 6250 B
2r
7. 7520 n
8. 1242
sr
7. 6172 B
6. 6043 B
4r
7. 7135 n
8.2308
9o*
_4r+40 +4J
7. 1862
7. 9072 B
3r"-(-40o+4J
7.1804
7. 8679 B
-2r+40 +4J
6.817
7. 456 n
r"+40 +4Jo
8. 4680 n
8. 8822
40 -f-4J
4.666
5. 807 B
8. 0913
8. 8270 n
9.2073
pjf^g +4J
8.7850
9. 8236 n
2T +40 +4Jo
8. 5144
9. 4910 n
3/ 1 +40o+4Jo
8. 3274
9. 3006 B
4r+40 +4J
8. 1627
9. 1494 n
57"*-f-4vQ~}~4^o
8. 0050
9. 0105 n
in'
4/~'-t-40 -4-3^
7.354 n
8. ]083
-3^+40 +3J
7. 5708 n
8. 2084
~~ ^ f ~f~40Q-f~3Jo
8.8838
9.0548-
40 -f-3Jo
4. 516 n
6. 2084
8. 5565 B
9. 218p
9. 5174 n
/ 1 +40 +3J
9. 2783 B
0. 2833
2/^+400 +3J
9. 0241n
9. 9635
3r+40 +3J
8. 8480 n
9.7850
4/^+400 +3J
8. 6916 B
9.6434
5r+40 +3J
8.5401,,
a 5128
m/a
m/J
77l /2 , 77J- 7
m'\ m'
m'
m'
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE F (L VI) Continued.
37
Logarithmic.
w I
Unit- 1 radian.
Cos
^
w-J
..;
10
.
TUB'
-4T ~ + ^o
7.7640
7.8364,
/w /
-sr f 4
7.4203
8. 3915
7. 8104,
8.6268
- r f- J,
8. 0479,
8.8018
4.518,
5.886,
5.70,
r +
7. 1339
7.8500
7.8421
a 4293,
sr + A
7.9669
a 6796,
4T + Jo
7.9760
a 7576,
*"
_4r+40 +2J
6.9002
7.6938,
-3r+4^o+2J
7.1638
7.8502,
I 2 f+4J7+2J
8.1860,
a 4016
49 +2^
3.76
e-oeoSn
8. 4157
8. 9760,
9. 1661
/'+4^o+2J
9.1714
0.1382,
2f-(-4fl -)-2J- ,
8.9358
9.8333,
3y-j_4# -j-2J
8. 7718
9. 6681 B
4/^-j-4^ -|-2^
8.6236
9.5372,
***'
j.n
176
5. 7516
4.7
r
7.8677
8. 6727,
2f
7. 8610,
8.2228
sr
8. 1026,
8.7296
4r
8.1538,
8.8728
f
r
7.9418,
a 7337
2f
7.9312,
a 7154
sr
7.7920,
8.6154
4r
7.639,
a 5001
f ;.' ' ::
f
_4f-(-4fl -)-3 < / _j
7.446
8.1156,
-Sr+^o+SJo-J',,
7.1858
7. 8677 B
- r+4+3Jo-^o
7. 6176,
7.9693
4^ -|-3J 2 a
4.804,
7.168
7.9368,
a 3724
/-(-4^ -j.3J ^"
7.7887
8.8492 B
2r +4^ -j-3J .J
7.448
a 4531,
3r-j-4tf -i-3J ^
7. 19J6
8.2026,
4r +4^o+3J -^o
6.978
7.9963,
**
2^ +2J
5.418,
6.292
7. 4754,
a 6636
6/? +6J
5. 418,
6.292
8.6328,
9.4351
1J0 2 !/
26 + ^
5.885
6. 719,
8.5059
9- 2804,
wV
2C +3J
4.974
5.896,
8. 0326,
8^ 1975
r ">X
6 +5J
5.935
6.780,
9. 2774
0.0330,
2^o
5.744,
6.535
8. 3811,
9.1030
r /0 j;' 2
20 +2J
5.441,
6.327
a 0917
8.6300
'So 'j' 2
66 +4J
5. 919.,
6.744
9.4432,
0.1464
1 ! /3
2^o+ ^o
5.301
6.14.9,
8.2302
9. 0152,
'"
6C +3J
5.301
6. 149,
9.1294
9. 7729,
2$ +2J
8.5904
9.3492,
9.8022
/'Jo
2^ + J 2<t
4.502 n
5.41
a 1011,
8.8726
6^ +5J ^"
4.502,
5.41
8.0554,
8.9263
*3 _/
J i
2^ + J
8.5592,
9. 3245
f i
26 +2J 2 a
4.057
5.021,
6.887
8.1804,
f i
6<? +4J -v
4.057
5.021,
8. 2718
9. 1021,
^'9j^ coe Arg.
where C represents the coefficient.
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
Logarithmic.
TABLB G (LVII).
S sin </>+C coa if/
Unlt-1".
Cos
to-
te-*
M
.
1C
.
<1> 5r+20 +2 J
8.81
1.082 B
1. 5710
1. 612 B
<l> 4f +20 +2J
9.009
1.5493
0. 9,89
<j> 3/ > +20o+2J
9.318
0^931
1. 604 B
1.916
<A 2.r+20o+2J 8
9.207
1. 6478
2. 1070 B
2. 2333
<fr- r+20 +2J
9.711
1.950
2. 3426 B
2. 3713
<& +20 +2J
9.196
2. 171 2 n
2. 5678
2. 565 B
9. 230 n
2. 3541 n
3. 1493
3. 7107 B
^+27" I +20 +2J
9. 220 n
1. 9114 n
2. 6867
3. 1657 n
^+3r+20 +2J
^+4/ I +20 +2J
a494n
1. 5372 B
1.2544,,
2. 3831
2. 1315
2. 8623 B
2. 6333 B
^+5/ 1 +20 +2J
9.100 B
1.018 B
1.9034
2. 4248 n
to
^-5r+40 +4J
9.771 B
1.042 B
.1. 868
2. 357 B
d> 4/'+40 +4J
0. O64. n
1. 723 n
2. 3515
2. 6814 n
^ 3r+40 +4J
0. 3185 n
2. 1626 n
2. 6961
2. 921 4 B
tj 2/ 1 +40 +4Jo
0. 497 B
2. 7787 B
3.0649
3. 0993,,
#- r+40 +4J
1. 0286
3. 2379 n
3. 1223
3. 9385 B
# +40 +4J
9.199
9.04 B
2. 6172
3. 2511 n
3. 4930
0+ r+40 +4J 8
0.7226
3. 1702
4. 1580 B
4. 9365
<&+2r+40 +4J 8
0.669
2. 7877
3. 7083 B
4. 3605
i+3f +40 +4J
0.9435
2. 5117
3. 426L,
4.0450
^+4r+40 +4J
~ j
0. 5122
2. 2732
3. 2042 B
to
<fi-5r
9.814
1.925
2. 634
2.984
<!>-r
0. 0434 B
2. 0527
2. 6896 B
2. 9432
<bsr
0. 3541 B
2.145
2. 675 B
2.744
d> 2/ 1
9.140
0. 362 B
2. 1351
2. 3850 B
2.4864n
<j> F
0. 4164 n
2.3504 B
3. 0929
3. 5397 B
|
9. 274 n
0. 1436 B
0. 3102 n
2.497
3. 1875 B
3. 5978
^+2.T
9. 137 B
9.918
1.9006,,
1.0453
2.8834
J+ttf"
9.465
0. 8 / 12 n
2. 5218 B
3. 3564
^+4jT
9.20 n
1.406 n
1.729
if
5r"+45o+3Jo
9.476
1.327
L889 B
2. 2299
4/ I +40 +3J
9.781
1.447
2. 1506 B
2.5419
^ 3.T+40 +3J
9.811
2. 1070
2. 6309 n
2. 8J508
^ 2.T+40 +3 J
0.3489
2. 5095
2. 9557 B
3. 0952
<]>- r+40 +3J 8
0. 9511
3. 3599
2. 7758
3. 9726
<!> +40 +3J
8.76 B
0.158
2. 7932 B
3. 3085
3. 4526 B
</>+ r+40 +3J
9. 961,,
3. 3609 n
4.3114
5. 0691 n
^+2/'+40 +3J 8
491
2. 9943,,
3. 8728
4. 4922 B
^-j-3/ 1 +40o+3J
1.' 0464 B
2. 7293,,
3.6067
4. 1945 B
^+4f+40 +3J
0. 678 n
2. 4992 n
3. 3946
Tf
V ~~ 5j ~T~ Jn
9.848
2. 0766 n
2.712
2. 9697 n
V^4l "4~ Jn
0. 0792
2. 1609 n
2. 6968
2. 7976 B
<j>zr + J
0.3941
2. 157 n
2.491
1.51
<j>zr + J
9. 013 n
0.248
2. 0455
2. 7898^
3. 2380
<ii r + Jo
9.901
2. 58,4
3. 2539 n
3. 6434
1
9. 885 n
0. 8,518
^+ /" + Jo
0.1664
1.836
2.448
3. 3029 B
^+2r + J
9.009
9.76 B
2. 1633
2. 6170 n
2. 2433
^+sr + J
9.38 B
2. 1064
2. 7194 B
2. 9212
^+4r + J
1. 9892
2. 6870 B
V
^-5r+60 +6J
2.3144
2. 9730 B
<l> 4/ 1 +60 +6J
2. 9538
3. 3785 n
^ 3r t +60 +6J
3. 3102
3. 5843 n
y~2/ 1 +60 +6J
3. 4970
3. 8423 B
^ ^+60o+6J
3.9455
3. 7269 n
^ +60 +6J
9.95 n
1. 1109 n
3. 1673 B
3. 9296
4. 3377,,
Y~\~ ^ 1 +60o+6J
3. 9144 n
5. 0372
^+2f+60o+6J
3. 5594 B
4. 5942
^+3r+60 +6J
3. 3121 B
4. 3236
to 3
^-5r+20 +2J
2. 1657
2. 7221 B
d> 4/ 1 +20 +2J
2. 1255
2. 8004 B
^ 3r+20o+2J
2.234
3. 1304 n
<!> 2r+20 +2Jo
2.576
3. 3804 n
<1> ^"+200+2^0
3. 1995
3. 8325 n
ip +20 +2J
0. 344 n
1.017
2. 689 n
3. 4822
3. 9938 n
<l>-\- / 1 +20o+2J
2. 2480
3. 2839 n
^+2r+20 +2J
9.45
3. 1612
3. 8424 n
No. 8.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
39
Logarithmic.
G (LVII) Continued.
S sin <j>+C coe <j>
Unlt-l".
Cos
^
^
,H
;
1C
-
,
f_5r-20 -2J
2.700,
15481
TO
\_ip 20 2J
2. 817,
3. 6251
V_or- 20 2J
2. 9247,
3.6905
i 2.T 20 2J
9.59,
3. 0241 B
17470
<i F 20 2J
3. 1364,
3.8346
i 20 24
0. 117 0. 95,
2.297,
2.7856,
3.6614
i+ F 20 2A
2.8942,
3.5604
(4+2.T 20 2J
2.297,
11129
if
<i 5F+60 +5J
2.4885,
11691
vo T
<A 4/ f +60+5J
2.976,
15560
5 3r+60 +5J
3.6541,
18829
(4 2/"+60 +5J
3.9514,
4.1632
/^+60A+5Jg
4.3903,
4.0037,
L'' +60/\+5Jo
0.295
1.366
3.6364
4.3301^
4.6662
$+ f+60 +5J
4.4005
5.4966,
<4+2/ 1 +60 +5J
4.0582
5.0612,
^+3r+60+5Jo
3.8204
4.8027,
.
v 5 "T 2^o < ^o
2.426,
10684
^ 4/^_i_2^ 4-J
2. 399,
3. 0310
^ 3/^-4-2$ +^
2. 410,
3.1305
5--2.T-i-20 +J
2.701,
14602
y~~ ' i 2vQ~T~ o
3.2842,
3.8558
V *i *^o~i o
0.444
1.188,
3.0569
3,7266,
4.1122
> i r^ I Oa i ^
2.8541
3.5823,
^+2r+2flJ+4
3. 2191,
17635
,
A-5r-20 -J a
3.1551
19530,
<l>4r26 J
3.2454
3.9948,
A 3f 20. 4.
3.3100
4.0023,
ibir 20 j
9.93
3.3277
3.9401,
ij /" 20 J
3. 1976
3. 4598,
A 20 J
0.490,
1.324
10145,
3.7326
4.2787
i+ f 20 J
3.3632
3.9402,
^+2r-20 -j
2.7792
15224,
fcf /
,j_5/-+20 +34,
2.2738,
2.847
V 1 ~~"4j ~|~Urt~T~"^O
2.116,
3.0290
tj 3/^+2^0 -j- 3^
2.5858,
3. 3787
V ~~2/ ~y~toVft~7~O^Q
2.809,
3.5429
A / ? -|-2^ -i-3J
2.650,
17297
W ~T~"0~1 "^0
9.98
0. 60 n
2.873,
aess
17980
cj+ /'+20 +3J
3.5126,
4.2856
#+2r+20 +3J
9.46,
13438,
4.1208
,/J
w~~df ~f~Ov/*T~4d()
L9950
2.7422,
~"~4y ~4~6w(|~j~4dn
2.6112
3. 1949,
d 3.T +60 +4J
3.0556
15583,
tf 2f+60 +4J
17934
17947,
^ / I +60 +4Jo
4.2260
4.4064
^ ^-60 +4J
9.98,
0.76,
3.5017,
4.1098
4.3552,
^+ /'+60 +4J
4.2852,
5.3521
#+2r+60 +4J
3.9567,
4.9249
*
tJ-5r+20 e +2J
2.5018
10963,
# 4/'+20 +2J
2.453
10935,
d> 3/"+20 +2J
2.4799
3. 2779,
^ 2/'+20 +2J|)
a 9375
16294,
^ r"+20 +2J
12833
3.8982,
d> +20 +2J
0.025 B
0.60
2.634
3. 2781
4.0439,
$~T~ * t~2vn~\~2an
3.5607
4.2381,
^+2r+20 +2J
14629
4.1704,
1*
^_5r-2
3.0090,,
1 7477
^ 4f 20
3.0676,
17445
<f>zr2o a
3. 0764,
3.6664
A 2r28
2. 958-,
1 3121
<j> r26
3. 1140
4.0201,
r -20
0. 305 L 127,
2.912
3.5491,
3.9085
A-\- P 26 a
I
3. 0396,
3.6320
A+2F-26,
2.4706,
12330
40
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE G (LVII) Continued.
Logarithmic.
[Vol. XIV.
Unit-l".
Cos
tu-
w-J
w->
u,o
to
w>
f
^-5r+66' l0 +5^ -J'
2.006
2. 7505 n
^-4r+60 +5^ -j
2.335
2. 981,
V 3r+60 +5<l -.r o
2.544
3. 1436 B
<!- 2r+60 +5J -.
2.718
3. 2445 n
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3. 9420
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2. 3824 n
3. 4488
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9,6
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2.911
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3.3047
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3. 6294
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3. 3406 n
3. 9330
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0. 5910
3. 1266
3. 8021
4. 1894
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3. 0472
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1.085
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0.35
1. 895 n
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1.463
2. 5146 B
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2.064
3. 0255 n
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2. 6816
3. 6280 n
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9.04
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2.636
3. 3284 n
3. 7399
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3. 0572 n
3. 6430
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2. 9121 n
3.5491
V.
^+ 40 +44
0. 775
1.65 n
3. 1052 n
3. 0342 n
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1.10
3. 1888
3. 6104 n
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0.65
1.54 n
3.7520
4. 5812
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4. 3244 n
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1. 260 n
2.081
3. 1240
4. 1388
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1.005
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3. 5356 n
3. 3560
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1.228
2.093
4. 3980 n
5. 1827
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4. 1155
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1.106
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2.831
4. 1803 n
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1. 146 n
1.88
3. 0422
4.0180
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1.321
2. 152 n
4. 5658
5. 3010 n
v
0+ 45 +3A
3. 8375
4. 0446 n
#- 40 - 4,
3. 2197
3. 9650 n
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4. 2553 n
4. 9349
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3.0024
3. 8634 n
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9.98
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2. 956 n
3. 8331
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3. 0757
3. 975? n
</>+ 40 +4J
0.46 n
1.32
3. 8514 n
4. 6436
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0+ 40 +4J -J
2.442
1. 846 B
#- 40 -2J +2-
3. 2486
4. 0585 n
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3. 2818 n
4. 1441
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1.15*
3.9421
4. 6972 B
S sin t+C cos ^='SCw*riPri'Qj 2 t cos Arg.
where C represents the coefficient.
H. TABLES FOR THE DETERMINATION OF THE PERTURBATIONS OF THE
HECUBA GROUP OF MINOR PLANETS.
DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS FOR W AND FOR THE THIRD COORDINATE.
It would be futile to attempt to give a brief but comprehensive outline of the fundamental
developments in the theory of Bohlin-v. Zeipel which would assist the reader to an understanding
of the construction of the tables. In broad outlines, the problem is the integration of Hansen's
differential equations for nSz, v, and -> by means of the method developed by Bohlin and
according to the modifications introduced by v. Zeipel for purposes of numerical computation.
The first division of the problem is the development of functions of the partial derivatives of
the perturbative function; the second division of the problem is the integration of the Hansen
equations in the form of infinite series.
For the theory the reader is referred to the original works of Hansen 1 , Bohlin 2 , and v. Zeipel*.
As indicated in the introduction to the first section, unless otherwise stated, the references to
Bohlin refer to the French edition and are designated by B; references to v. Zeipel are desig-
nated by Z. Although duplication of material which can be found in either reference is to be
avoided, our experience in attempting to reproduce v. Zeipel's tables led us to fill in certain
gaps which are troublesome to the reader and the computer.
The first section of v. Zeipel's theory is concerned with an independent development of
Hansen's differential equations for ntiz and v and a repetition of the differential equation for
*t and the introduction of Bohlin's argument 6. In passing, it is well to emphasize two
cos t-
facts: First, the variables e and /"used throughout the theory are analogous to Hansen's e and/;
the dash is unnecessary, for the physically real values do not appear. Second, the constant
elements a, e,n,c, Q,,i are neither osculating nor mean elements; they are defined in the section
on constants of integration.
The perturbative function and its partial derivatives are developed in Fourier's series, in
which the arguments depend upon the relative positions of the disturbed and disturbing
bodies and in which the coefficients are infinite series in ascending powers of the eccentricities
and the inclination of the orbits. The coefficients in the latter are elliptic integrals depending
upon the ratio of the semi-major axes.
Since these elliptic integrals are functions of the ratio of the semi-major axes, or of the
mean daily motions, they can be .developed in Taylor's series, in which the given function and
its successive partial derivatives are expressed for exact commensurability and the series pro-
ceeds according to a small quantity w, defined by w=l 2 ft, where ft is the ratio of Jupiter's
mean motion to that of the planet and where ft differs but little from These elliptic integrals
enter the coefficients in all of the subsequent trigonometric series. Hence all the coefficients are
series in w. With some exceptions the terms in w, w, and v? have been used. The develop-
ment of all functions in powers of w is the essential principle underlying the group method of
determining perturbations.
The following pages contain the tables which are, in general, parallel to those of v. Zeipel.
At the end of sections 2, 3, 4, 5 there are brief written comparisons. To facilitate comparisons
' Auseinandersetrung einer tweckmassigen Methode HIT Berechnung der absoluten StSrungen der kleinen Planeten.
' Fonneln un<J Tafeln cai gruppenweisen Berecknung der allgemeinen Storungen benachbarter Planeten. Nova Acta Reg. Soc. Sc. t'psalienslx,
Set. Ill, Band XVII, 1S96.
Bur le Diveloppemcnt des Pertabations Flane'taires. Application aui Petites I'lanetes. Stockholm, 1902.
' Angenaherte Jupiterrtcrongen fur die Hecuba-Gruppe. St. Pitersbourg, 1902.
41
42 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [VOLXIV.
with v. Zeipel's tables, those numerical quantities which are in disagreement are inclosed in
brackets. There are also certain mathematical developments useful to the reader. These
relations are sometimes taken from v. Zeipel and sometimes supplement his text.
Certain simple functions of the elliptic integrals y t m ' n , defined by Z 19, eqs. (73), (74), (75),
are tabulated in Table I (cf. Z 23).
Tables II-IVw 2 (cf. Z 26-32), giving the partial derivatives of the perturbative function,
are computed according to Z 24, eq. (77), by means of Table I and B 184, Tables XVI-XVIII
and B (Ger.) 182, Tables XII-XIV.
The elimination of Jupiter's mean anomaly from the argument gives Z 25, eq. (78), in
which the coefficients are derived from Table II-IV v? by the formulae given in B 61. These
coefficients are tabulated in Tables V-VII W? (cf. Z 33-39).
fan* i;ilal-i[ '_.-* [:! )':> i>. rM iui viii 1o p.fiu'M'i v.- Unit ,j ,z6;i iui i'.>.\ni ::. /
1., Mis.: ,..'>/.;> !.M:>ii>Tliiii JIM i\i\tr iMPttmlSUft
urt UM[|/U;i>- !n ; Jicn-r'u> m* to .miJUoqsi ^ ba;;
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oil)
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No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
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MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
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MEMOIRS NATIONAL ACADEMY OF SCIENCES.
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No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
47
**
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48
MEMOIRS NATIONAL ACADEMY OP SCIENCES.
[Vol. XIV.
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No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
49
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50
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
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No. 3.]
MINOR PLANETS-LEUSCHNER, CLANCY, LEVY.
51
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MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
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NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
Logarithmic. TABLE IV.
53
Unlt-l".
n
1
2
3
4
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1. 898161 n
1. 898161
2. 283141 B
2. 283141
2. 153770 B
2. 153770
2.006171 B
2. 006171
1. 847875 B
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1.68250 B
1.68250
1. 51210 B
L 51210
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2. 683079
2.528880
2. 366191
2. 197675
2.02490
1. 84887
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2. 522558
2. 937464
2. 958044
2. 924776
2.858077
2.76886
2. 66352
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2. 522558 n
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2. 528880 B
2. 366191 B
2. 197675 B
2. 02490 B
1. 84887 B
n 1. n 1'
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2. 522558 B
2. 937464 n
2. 958044 n
2. 924776 B
2. 858077 B
2. 76886 B
2. 66352,
^0-1
n. n+2]+j!
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2. 812563 B
3. 024413 B
2. 794447 B
2. 523495 B
2.197675,
1. 76164 B
0. 74650 n
Rtt-t
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n. nj if
2. 522558 B
2. 522558
3. 024413 B
2. 462558
3. 07659,8 n
1. 724281
3. 058902 n
1. 856833 B
3. 001314 n
2. 093957 n
i 12968 B
2.81684 n
2. 09513 n
RO-I
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2. 812563
3. 261391
3. 246209
3. 190606
3. 108845
3.00884
2. 89541
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3. 49405 B
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3. 36728,,
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3. 70912
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3. 3037,0,,
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3. 30370 B
R-i
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3. 81842 B
RO-I
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3. 21895
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2. 72638 B
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2. 72638
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2. 94112
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2. 72638
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2. 72638 B
RO-O
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2. 51524
2. 84832
2. 78195
2. 69286
2. 58759
2. 47019
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2. 51524 B
2. 84832 B
2. 78195,
2. 69286 B
2. 58759 B
2. 47019 B
1 '0
n+l.-n+I
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3. 25180 B
3. 45486 B
3. 34235 n
3. 21906 B
3. 08741,
2. 94903 B
75
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n 1. n+1
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i 25180 B
3. 62946 B
3. 66471 n
3. 66409 B
3. 63529 B
3. 58453,
R
n+l.-n-l
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3. 25180
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b
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3
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No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
61
m
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CO
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4 47
lO CO
Hi
rHIO
1 4
i-H
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CO kO
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4 4T
1
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l
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4 41
74
CO CO OO
CO ?0 ^*
CO O ^*
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1
4 4 1
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ci coo
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->-CO r-( Q ^H
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1
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o* 0*0*
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62
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE VII.
[Vol. XIV.
Unit-l".
n
l
2
3
4
5
e
fi (n _n+l)+r
/
- 79. 10
- 191. 93
- 142. 48
- 101. 43
- 70. 45
- 48. 14
- 32.52
fl.(n.-n-l)-7
S
+ 79. 10
+ 191. 93
+ 142. 48
+ 101. 43
+ 70. 45
+ 48. 14
+ 32.52
JZ,. 'n+l.-n+l
~\~n
+ 372. 6
+ 482.0
+ 266.7
+ 130.9
+ 52.0
+ 9.6
- 10.7
~f~7t
+ 293. 5
+ 865. 9
+ 979. 2
+ 942.4
+ 826. 9
+ 683. 6
+542.1
.R.. (n+l. n 1
TT 7
- 293. 5
- 290.1
- 124.2
- 29.5
+ 18.5
+ 38.5
+ 43.2
7Z,. (n-l.-n-l
-'
- 372. 6
- 1057.8
- 1121.6
- 1043.8
- 897. 4
- 73L7
-574. 6
Bo.,(n.-n+2)4V
- 649.5
- 1057.8
- 622. 9
- 333.8
- 157. 6
- 57.8
-5.6
/Z . I (n.-rt4V
- 333. 1
- 1057.8
- 1192.9
- 1145.3
- 1003.0
- 828.
-655. 9
+ 333. 1
+ 290.1
+ 53.0
- 71.9
- 124. 1
- 134.8
-124.5
.Ro.j(n.-n-2)-7r /
+ 649.5
+ 1825.5
+ 1762.8
+ 1551.0
+ 1284.8
+ 1020.6
+786.
Bj. (n.-n+l)+i
r"
- 3119
iJ 2 . (n-2.-n+l
14V
- 2330
JZ,., n-l.-n+2)4V
+ 5118
JZ,., n+l. n)+a
^
+ 2012
lZ|.j n 1. n)+7i
J
+ 2012
1Z,., n-l.-n)-^
- 2012
J?J.J(TI~~I. ft) 3
S
- 2012
/I0.j(7l. ""Tl"^! )~7~7I
{
- 6583
JZ .i(n.-n+l)-K'
+ 1656
~r 1
IT" +
-7?o-o tt~l.~tt)""* C
'+**
- 533
-Ko-o *M~1- ^)~H
>4V
+ 533
/J ., ) n-l.-n+2)-<5+7r /
+ 873
Ro-o n+l. n)+c- V
.RO.O n 1. n) d n'
+ 533
- 533
/Z . (n.-n+l)+r,
f
+ 327.5
+ 705. 2
+ 605.3
+ 493.0
+ 386. 9
+ 295. 2
Ro-o(-- n -l)-^ /
- 327. 5
- 705. 2
- 605.3
- 493.0
- 386. 9
- 295. 2
U ( n +l n+1
4V
- 1950
- 2850
- 1897
- 1163
- 643
- 299
tfl;(n-L-n+l
+* /
- 1622
- 4260
- 4923
- 5107
- 4898
- 4432
/Z,. (n+l. n 1
TT 7
+ 1622
+ 2145
+ 1292
+ 670
+ 256
+ 4
JR,. (n-l.-n-l
-Tt 7
+ 1950
+ 4966
+ 5529
+ 5600
+ 5285
+ 4727
#., n.-n+2)+^
+ 3096
+ 4966
+ 3410
+ 2149
+ 1223
+ 594
RQ.I 71. tl)-|-jr /
+ 1786
+ 4966
+ 5831
+ 6093
+ 5866
+ 5318
/J 0>1 n. n) itf
- 1786
- 2145
- 989
- 177
+ 325
+ 587
RQ.I ft. ft 2) 7
/
- 3096
- 7786
- 8252
- 8065
- 7413
- 6499
8
fl 3 . (n.-n+l)+j
[
+23018
|
JZ 3 . (n-2.-n+l
14V
+15418
(2
RL n-l.-n+2)4V
-33562
JZ,. n+l.-n +7
!*
-13734
JZ-i. n 1. n +7
t*
-13734
JZi. n+l. n 7
r 7
+13734
JZ,. n 1. n 7
/
+13734
.Ro.2(n. n+l)+7
I
+40061
feS
Rf,.^(n. n+l) 7
I
-11862
"0*0(1 1. n) e
r+TC 7
+ 3280
+ f
/Zo.o(n+l. n)+i
I+;t/
- 3280
R t . a (n 1. n+2
- 5854
JZ . (n+l. TI)+
r It'
- 3280
^ i-2 ""*
Ro-o( n ~l- n) <
t-V
+ 3280
ta *"* ^
R . (n. n+l)+7
r 7
- 1303
- 1127
/Z . (n.-n-l)-j
r*
+ 1303
+ 1127
+ 838
B,. n+l. n+l
4V
<
fi,! n-l!-n+l
+13600
P
/f,. n+l.-n-l
-TT 7
- 7080
-14465
2
^,.0 n-1. n 1
-T/
-12290
1
,Ro.].(n.-n+2)+;
?
- 7475
*^0 ' 1 x ~~ ^/ "l ^*
-14430
I\Q.^\Tl. ~~ft) ^~7t
+ 4532
R^n.-n^)-,
C 7
+ 7475
+20370
NO. 8.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 63
With these tables we compute terms of the first order in the mass in Hansen's differential
equations for the function W and the perturbation in the third coordinate. See Z 7, eq. (33)
and Z 8, eq. (39). The first order parts of the equations are expressed in Z 41, eqs. (82), (83),
in the form of trigonometric series, hi which the coefficients are computed from the formulae
given hi B 67. These coefficients comprise Tables Vlll-XIVto 2 (cf. Z, 42-48).
Table XV (cf. Z 50, eq. (88)) is an auxiliary table of the same type of construction, which
is employed in the computation of terms of the second order hi the mass hi the differential
equation for W (cf. Z 53).
*
"
-
64
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
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MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
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o o
o o v ~o o
^~*~*~*~ ^S'^S
^H'^S'
^-i-t^^-^ OOOOO
-<
CO CO
CM CM CM CM
faq tc; fc) ID K5V?
co coco co WH^H
tq'ijjtq'^tq' kf* 3 ? bf'D'i^flD'lD 5
tejtej
tej tej fej tej
CM CN CN C^
tqlDtDU) K?b?
CO CO CO CO ^ "N
hi*
m, jo
wa.
w
H)
ta
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY. LEVY.
75
.
7
*S ?: : si
I
"V
-.
* S
3k
*^ *^1 J. -1 -'I 1
-1 ^> -"? * -^
C-J C^
H
CO CO
J5
u
^i 1 "T"
^* >5 39 P9 ^ ,
-,
ss
M 00
CO C5
+ 1
K : lr 7
+ 77
<
g
i
N V ~T~V s
vvv - - ++ 1 1 1
febb tits o ;q- to <a
"sT'sTI? Ztt "g^'e's^s"
III ++ 1 1 1 1 1
V V ++ 1 1 V
+1 ?^77 jkji
r-<-H gggS <^+|^
+i i i i i +~ i
S g ^- S g g S
7+1 1. 1. 7+ 1 + 1
SSSSS
e J. + 1 + 1 '. '. '. '.
oc oooo ^
L-'.--,-* i.?h? ^fcPhfcPh^k^PK-?
55 ^" ^S^
t n JOJDBJ
" J
^ -
^1
i
" II X -I
S? 1
++ i i
s
76
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV
s r
CO CO
CO ^^
O Oi
-I
t~
CO OS t~
O O5 Tf<
CM 00 CO
CO
+ +
com
CO CO CO
i 1 1 i r~
r-lin CO
CN
1 + +
CO CJ O 'O t-* CO
CO C^l CO *O *0* O5 1
f-i *^ c; ic co o
+ + 7
CO
OS COO4
t^ O CO
co i i u5
1 1 ^
+ +
t- a> co
Ci CO CM SNI
CM N CO 35
ii co 55 1-1
+ 1 +
"tf
o" ct r-. CN co -(
CO CO tft> C<J O OS
C$ b* OO
SOS OS
IMO
* rH iO
t~O
OS CO C
rH |- Ci
OO
r^ t^- co !> I s - t
OS i ( lO * 1 ^* r-
^s ^a g
s ..i-
3
n rHlS
+ +
CO QO OS
+ + 1
cc eocN ?i t^ oo
rH ^ CM CO CO O
CM COCO W CO ft
+ + 1
iO W CO
+ +
t* CO
CO O f""" CO OS
+ 1 + 1
C<I
C^CO^'CCO ^PTO O
+ + rt +7 +
fH
t* t- CO
ifS os co
+ i
0r-l C~.
^i-l 3lO CO
+ + II 1
CO
os r^ ?o r^ o *^ c-i
^tf CO CO CO CO O Cr
N rHCO CO
o
1
e e"?
! ' i-i r-i
. +
i 11 IrHrH i-H rt i H rH
rn^? "s? "5" sess e? cT ssss
t- '.-?'. '. '. '. '. t^-e '. - '. '-
7 CM , CXI i-i I-H r-l r-i , , i-H rH i r- i
'..+'. 1 +1 + 1 . '. '. + + 1
MM OOO NC*C>. OOC
rH r- 1 r-( rH
^^^ ^.^ ^ ^ ++ II ^ ^
^ + i-i TIM +^
fi SSS -C- .... ccc
rHrH (MTC-l f-HrHrHl-1
. + '. . + '. 1 + \ + I . '.
05 OQ'CQ'
oo ci c-t ~t M *HMM oeo occo
O i~ OO CMCMN __. OO
XI JOpB J
1
M
s
No. 3.]
SO -^ -^ n
t- O t~
* 1-1 o
00
r ~
S) rH V O CO
O cc r- ^
99 O
OS 10 O * l~-
OC O I s * O ^'
jg
O O ^^ O 'O
8
w l~^ o o^
W CM
+
+ + +
to cot- asm
r~ co i-i T
^1 !-!
1 ( I-H
; f? S 4
+ "+7
+ i +
s s s s
I 1 1
9^ J^*"
x*s. 1
s rt'rt' R a
+ + 1
s s s s
s s e 8 a
a o o o
o o o
o r o o a
tfcf coV
: P JOWBJ
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
13 If. 1 . ,;-..-LViM HIT TO fO 1 -".^O-TVii
77
fll ori)
ffi 'Ki'*^ ^d4 lo
78 MEMOIRS NATIONAL ACADEMY OF SCIENCES.
INTEGRATION OF THE DIFFERENTIAL EQUATION FOR W.
With the exception of Tables LVI and LVII all the following tables are concerned with
the integration of functions whose coefficients can be derived, more or less directly, from the-
preceding tables. The terms of first order in the mass, before and after integration, are of the
type , A
where . C v . q = C . p .g + C V p. q -w + C^p.g-w 2 H ---- (see Z 25)
and A = [n + r-%(n-s)]s + (n-s)6+i IL+i' II'
In the argument A the factor n is always a positive integer; the factors r, s, i, and i' are
Tc
positive and negative integers. Evidently, the factor of is -~ where k is any positive integer,
and the arguments in a series are I n I r I s A. Within the extent of Bohlin's tables all of the
coefficients can be written in symbolic form from B 188, XVII, XVIII. In the notation for
the coefficients the particular values of r and s are given, and there remains to be found only
the positive value of n, if there is one, for each multiple of -~-
||
The following tables present, in skeleton form, any series of the given type. There are
properly two tables, one for perturbations in the plane of the orbit, and the other for perturba-
tions perpendicular to the same. The headings A and I are defined by
J = n-n'
Considering first the tables referring to the plane of the orbit, omitting for the moment
the arguments bearing the subscripts 5 or <r, the argument A for any term is read from a
Ice
main heading -5- and the first two columns under this heading. The tabulated numbers are
the respective factors of 0, A, and I. The degree of the factors in the eccentricities is indicated
in the subscriptsy-g'in the symbol for the coefficient. Further, when j tt = 1
Hence the coefficient of A is also the number n in the proper table of the numerical values of
the coefficients. For instance, in the function T 2 (Z 41, eq. 82) we have for one term
where F, taken from Table VIII, is numerically
F,. (n-l. -)n_4=- 1514" + 5780"u>-8976'V.
Adding e </> to the argument and taking the coefficients from Table IX, we have also in the
function T, --
n-l.-n)^ sn 2e-
where G V9 (n- !.-)_= +452"- 1475"w+ 1451"^.
In this manner the series is built up.
The coefficients having subscripts d and a belong to terms depending upon the mutual
inclination of the orbit planes. They differ from the preceding type of terms in three ways.
In the first place the subscript signifies the addition of J and 2" to the argument, respec-
tively. Evidently, if J is added to the argument, the factor of A is not n but nl, from
which we determine n. Lastly, these terms contain the factor f, i. e., within the extent of
our tables the exponent t is not greater than unity.
For the tables referring to functions which concern the perturbations in the third coordi-
nate the same explanations hold, with the exception that the additional subscript rc f signifies
the addition of 11' to the argument.
These tables, in connection with the proper tables of numerical coefficients, enable the
computer to write a complete series by inspection or segregate any term of given degree and
given argument.
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
79
H
r-T
-
-- " ~ <O oo
:- 1OO ICO
N
|
!
*
h
"7
fc
*
n^ -r= _ oo^ o=^
*
B ^t- ^ ^^00 , _00
5
7
1
"
^ o o
?
- - -
a
"7
t
*
CO CH-, CO^ OV<0 0<0
*
1 1 1
N
7
"I
i
MO ecoc* -H IN
1
2
-.-. m m -.
.
2
7
"T"
-
*
*
4
"
<O f "" r-
e "cc
*
7 7
1
->
7, _ _
O M OT M M v M
N
"7 "77 "77
c?
a
n co o-roe on,- * _r-,, CHCO nt .a ^cs = ^=o n^. . N^^W^MW
C4= ^C O^X 0,,* 0X> OX,, NOO N 00 NNOCO r.00 <NOC0 NOO^eNO
H -
i 7 77
r
.O M o ^ o, ^ eo o r* ^ ec o o ^
1
o o w N o-* o , o * * * o o-^t o
^
7 "77 "77
+
-i
C4 O ^> v-ieo r.- eOi^iC OC4V *-":7i7. O^<OC t-OOCOt^ 74 ^-7J-^ :7 ^ r: : - ^- ^ O * CO * CO ~H C 1 *
*
7i O, O, 7) ' 717]' 7 i 71 "C OC4C4O,OOO,^>OCOO,,CCO,3OO,X,O, O,XO,O^<
''<
7 77 77
|
0^ ^ W K<* ***
o N cs o^ o^ o-* * o w c*ei<oe*eci<o wo MtDC4M M M ri M
1 j
1
~~^ IZ ^ ^,^. ^. **** * +^
J9 S-J -5- -s- ++H {? s- +n?e"S-e =?=! -S^il CCS? Islsc? ||p
I-J: B !=! *""" fcfi tttt tititlt ttttt ill Iltlit ttttttt
'. + i '.'.+'. i +i+7 '. i. >. +7+7 +7 i Lj '.7 7+7+7 '. '. '. '. '.7 '. 1 7 +77++77
5 55. -55. 5-5-5. 5555 555 55S5 55 5555 55555 555 5\5\5555 e e e e a e e
80
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
it
+ 1
c
a
E
m
2 +
s
o*
h
7
r
O O 04
r* O
_U
7
CO M
N
7 ~7
T -,
<M -f o -t- c-t -J r^eoeoua ci --o o -* c
i i
1
n
rH I-H
1 1
i
TJ M M rj-
-* *H co o on < co o
.
eo ec ^o
rt " >0 --""" *..5
N
7 7
+
CSO^ -HCC OV NO
- CO CO-,10 ONO(N- M U5NCN
1 1
1 II 1
- ,__, . *<
7
_
7 "*
o o
*
w
(N <N
N
^-. -.
1
cS ^
040 0^00 0^-^00 WWWO
1 1 "*"j
=
' 77
r -
^ o ** eo
O (N O <N
N O) -*
O O *J> ^ O *
I
7 "7
+ ^
;
.
O -V NMtO OO*WtO Of * OO
1
OOT,0-,^.00 0-.<=-00 00 OOO^T,
;
7 7
-
P4 ~ ~f o o * ri d<o
CM <N O)C>4^0 M d<O C4 C4 ~~ r > 7 1
I 1
Itz :
tev +ili \ -t ivilvl
+^ ++77 ivvA +i+7JL7
2 aeaa f? , i ST s 'aa'e
+ 1 1 1 1 1 frSc-l 1 + 1 1 ' 1
e a 'S^B''?'? a a S?QS^S' e'a'Ja 9'aa'?
1 1 1 1 1 1 1 1 + + ^ + e ^ 1 1 1 1 MM
\\ +7+7 "Ti +T c i'+f c i 1
+7+7+7+7 iTi.Tii +7+7 +7+7
^^ O^O^O ^i^^Ti^H '^O^o'^^^'*^
.^-5-5.5.-S. 5-5-S.S5-5. 5-S-S-S. S.5^.
-I
NO. s.] MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 81
Our problem is now the integration of the partial differential equations Z 7, eq. (33), Z 8,
eqs. (37) and (39), and Z 9, eq. (47 1 ).
In the trigonometric series to be integrated the argument is a function of 6, s, <f>, A, I.
The last two are constants. According to the principles of Hansen, <f> occurs outside the opera-
tion. Numerically, however, it is equal to s. The argument 6 contains e implicitly. See Z 9,
eq. (43). Hence we must, in general, write
F(e, d)
and
= _
ds ds 50 ds
In order to set up the partial differential equations from the total derivative, the following
notation is introduced:
F(t, 0)=[F(s, ff)] + F(t, 0)-[F(s, 6)]
where [F(e, 0)] signifies that part of the function which is independent of e. Again, since s has
the period of the planet, there can be no secular terms in e (with the exception of the function 0),
i. e.,
On the other hand, the argument varies much more slowly, and there may be secular terms
in 0. Hence
and may occur outside the sign of integration.
Owing to the presence of the required function in the differential equation, the integrations
must be performed rank by rank where rank is defined as follows:
In the course of the developments there arise negative powers of w. Since w is a small
quantity, these factors increase the numerical value of the terms, or, in other words, they lower
the order. Therefore, it is better to define order in terms of both the disturbing mass m' and w.
For this purpose v. Zeipel makes the assumption that both w and -^m' are quantities of the
first order. Order so defined is called " rank," and the word "order" is reserved as usual for
7/1 ' a
the powers of m'. The factors 5- are arranged according to rank in Z 53.
Any function is then written in the form
where the subscript denotes the term of lowest rank, for F { (s, 0) contains terms of more than
one rank since each coefficient is itself a Taylor's series in w. In assigning rank it is to be noted
that the coefficients in all the preceding tables contain the factor m' implicitly. The implicit
mass factor is indicated at the foot of each table which follows.
On the basis of the foregoing principles, the differential equation for W,
d_W == dW + bW d0 = T
ds ds d0 ds
expressed in Z 52, eq. (91), is broken up into four equations, Z 53, eqs. (95, 95 4 ), according
to rank, and before integration they are again subdivided according to parts which contain e
and parts which are independent of s. The total derivative is then in the form of eight equivalent
equations, and the integration can be performed in the following order:
W t ; F 2 -[FJ; [FJ; F 8 -[FJ; etc.
It is possible to avoid the computation of T 3 , as v. Zeipel did, by the introduction of some
auxiliary functions, but we found it preferable to tabulate them.
Employing Table XVo, and by inspection of Tables VIII, IX, X, XI, T t is written directly.
(Thas no terms of first rank.)
110379 22 - 6
82
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
TABLE XVc.
r.
TJnlt-l"
Sin
w
w
1C*
-J
- f+#
+ 43.1
- 43.1
- 128.0
+ 128.0
+ 171
- 171
+ 20+2J
2t-<j>+28+24
<l>+28+24
+ 271. 5
- 67.1
- 294.9
- 636.6
+ 108. 2
+ 740. 6
+ 526
+ 32
- 734
2t+ 48+44
3t-<l>+48+44
t+<l>+40+4J
+ 159.9
- 45.7
- 167.4
- 593.3
+ 122. 1
+ 637.4
+ 869
- 174
- 984
2i+<{>+68+64
- 81.7
+ 418. 9
- 907
9
28+24
i-<l>+28+24
- t+<l>+28+24
-1180
+ 273
+1496
+2962
- 179
-4265
- 2935
- 1170
+ 5572
9
2t-<!>
9
- 173
- 211
+ 384
+ 512
+ 899
-1410
- 684
- 1921
+ 2605
n
e+ 48+44
2 t -<l>+48+44
f+48+44
-1514
+ 452
+1679
+5780
-1475
-6656
- 8976
+ 1451
+11172
n
2t+ 28+24
St-<l>+28+24
c+<j>+20+24
- 6
- 83
+ 136
+ 408
+ 262
- 878
- 1307
- 564
+ 2285
?
2i+ 60+64
3c-<l>+68+64
,+<i,+68+64
-1149
+ 360
+1227
+5902
-1734
-6415
-12820
+ 3301
+14400
^
2 t +</>+48+44
- 102
+ 112
i
2t+<l>+&8+84
+ 750
-4900
5'
28+ 4
t-<!<+26+ A
- c +<l>+28+ 4
+ 318
+ 222
- 646
-1081
-1012
+2452
+ 1552
+ 2227
- 4296
1
t+ ^
2t-<{>+ A
<i>+ *
+ 130
+ 112
- 285
- 484
- 565
+1211
+ 808
+ 1393
- 2475
n'
t+ 48+3J
2e-<l>+48+3J
$+48+34
+2279
- 580
-2460
-7160
+1410
+8138
+ 8896
- 520
-11342
n'
2t+ 28+34
3t-<!>+28+34
t+<f>+28+34
- 314
+ 127
+ 291
+ 702
- 399
- 537
- 90
+ 598
- 478
Y
2t+ 68+54
3t- </>+68+54
t+<!>+68+54
+1887
- 542
-1974
-8417
+2221
+9002
+15550
- 3377
-17350
I
i
2e+<{'+48+54
- .".'," >..,;..". stirp')
Zt+j+W+U
+ 390
-1263
-1556
+7397
9
.-#
- t+<!>
+ 568
- 568
- 3106
+ 3106
*
48+44
,-<f>+46+44
- i+<I>+48+44
+6716
-2114
-7960
- 26627
+ 6488
+ 33462
+ 44700
P
t+ 28+24
2t-<f>+28+24
<{>+28+24
/, 4- 128
+ 535
- 978
- 3166
- 2505
+ 7431
- 23105
m'
No. 8.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE XVe Continued.
Unlt-l"
Sin
V
V
w>
1*
+ 69+64
2-#+69+64
#+69+64
+ 7969
- 2624
- 8819
- 41736
+ 12577
+ 47347
-111337
'/'
- + 25+24
-#+20+24
-2t+#+20+24
+ 2246
- 396
- 3596
- 6168
- 1494
+ 12561
+ 9351
1*
2t
3t-#
+#
+ 423
+ 357
- 780
- 1797
- 2207
+ 4005
f
2e+ 49+44
3t-#+40+44
+#+49+4J
- 1783
+ 924
+ 1220
+ 3946
- 3327
^ 1026
1*
2t+ 80+84
3 t -#+89+84
+#+80+8J
+ 6749
- 2247
- 7252
- 44127
+ 14052
+ 48051
If'
4
*-#+ 4
- +</>+ A
- 285
- 1004
+ 1574
+ 1210
+ 5771
- 8192
- 2475
If*
49+34
t-#+49+34'
- *+#+40+3J
-17218
+ 4253
+20345
+ 56961
- 8340
- 73031
- 79400
fir
+ 29+ J
2e-#+2fl+ A
#+20+ 4
- 1429
- 523
+ 2280
+ 6138
+ 3792
- 11302
+ 28347
if
t+ 2fl+34
2t-j+2d+3J
#+20+34
+ 1725
- 1003
- 1492
- 3054
+ 3753
+ 677
+ 13097
v
+ 65+5J
2t-#+6+5J
#+60+5J
-23773
+ 7038
+25974
+108605
- 28427
-122380
+251019
>? ^
- t+ 29+ 4
-#+29+ 4
-2+#+20+ J
- 965
- 2068
+ 3785
+ 3533
+ 10582
- 16928
+ 39011
v
2t+ A
3-#+ 4
+#+ 4
- 820
- 470
+ 1488
+ 3797
+ 3185
- 7870
||T
2t+ 49+34
3t- #+49+34
+#+49+34
+ 1815
- 1181
- 853
- 1190
+ 3807
- 3161
--?'
2t+ 49+54
3t-#+49+54
+#+49+54
+ 4294
- 1571
- 4414
- 17092
+ 6629
+ 17198
^
2+ 89+74
3t-#+89+74
+#+89+74
-21544
+ 6700
+22868
+126397
- 37167
-136294
I"
-#
- .+#
+ 866
- 866
- 4261
+ 4261
*
49+24
-#+49+24
- +#+49+24
+10682
- 1815
-12428
- 28347
+ 474
+ 37322
+ 32120
>!"
+ 29+24
2 -#+29+24
#+29+24
- 1498
+ 1136
+ 861
+ 450
- 4394
+ 3794
- 22127
m'
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
XVc Continued.
[Vol. XIV.
Unit-l"
Sin
w
w
1*
+ 60+44
+17790
- 69344
ttttll-
2-0+60+44
0+60+44
- 4675
-19046
+ 15200
+ 77260
-135954
9"
-0+20
+ 1634
- 7081
+ 16199
f Ti-il I
-2+0+20
- 1634
+ 7081
,"
2+ 24
+ 328
^ 1710
3-0+ 24
+ 154
- 1141
+0+ 24
- 591
+ 3420
1
,/
2+ 40+44
- 5879
+ 19019
3-0+40+44
+ 2032
- 7361
t+0+40+44
+ 5807
- 17998
,/J
2+ 80+64
+17340
- 90064
3-0+80+64
- 5018
+ 24266
+0+80+64
-18102
+ 95820
jJ
-0
i- 866
+ 4260
<',"!.' -
- * +
+ 866
- 4260
J 3
40+34-2
+ 609
- 2958
+ 6763
-0+40+34-2"
+ 232
- 1656
tWfV -
- +0+40+34-2
- 1044
+ 5600
s ;
> a
+ 20+24
- 1760
+ 71S9
2 S -0+20+24
- 331
+ 3096
0+20+24
+ 2677
- 12681
+ 30930
; 2
+ 60+54-2"
+ 578
- 3543
2 -0+60+54-2"
+ 10
- 299
0+60+54-2
- 780
+ 5023
- 15302
;'
-0+20+4-2
-2f +0+20+4 -2"
+ 866
- 866
- 4260
+ 4260
+ 10988
f
2 + 4+2
+ 1152
- 4231
3 7ft iti
+0+ 4+2
+ 98
- 1634
1440
+ 7081
a
2+ 40+44
- 1795
+ 9459
3-0+40+44
+ 164
- 17
+0+40+44
+ 2229
- 12595
j
2 + 80+74-2
+ 392
- 2914
V ,'
3-0+80+74-2
- 40
+ 194
+0+80+74-2
- 482
+ 3691
*+ 0+ 4
+ 47.1
- 149. 3
+ 186
f-0+ 0+ 4
+ 27.5
- 111.4
+ 207
-*+0 + 0+ 4
- 90.4
+ 310. 5
- 455
?+ 30+34
+ 216. 1
- 655. 2
+ 749
|s-0+30+34
- 58.9
+ 150.7
- 93
$+0+30+34
- 229. 3
+ 722. 8
- 905
| + 50+54
+ 113. 8
- 499. 2
+ 892
^-0+50+54
- 33.5
+ 137. 7
- 213
fs+0+50+54
- 118. 2
+ 527. 9
- 977
^+ 70+74
+ 54.1
- 310. 2
+ 757
$-0+70+74
- 16.5
+ 91.3
- 209
+0+70+74
- 55.7
+ 322. 3
- 801
$+ 90+94
+ 24.5
- 173. 5
+ 537
-^-0+90+94
7.6
+ 52.7
- 157
$+0+90+94
- 25.1
+ 178. 5
- 559
m'
No. 3.]
MINOR PLANETS LEUSCHNER, CLANCY. LEVY.
XVc Continued .
85
Unit-l*
Sin
..
.
<c*
'
_|:^I^+3J
-1497
+ 419
+1729
+ 4732
- 966
- 5826
- 5963
+ 131
+ 8424
1
-i 4- 0+ 4
JE-<H- 0+ A
-|E+^+ 0+ A
- 114
- 224
+ 436
+ 385
+ 1006
- 1726
- 548
- 2186
+ 3220
\s+ 0+ A
JE + VH- 0+ A
- 208
- 55
+ 314
+ 781
+ 349
- 1316
- 1315
- 1026
+ 2641
TJ
?E+ 50+54
-1366
+ 420
+1480
+ 6114
- 1711
- 6793
-113C3
+ 2548
+13254
1
$E+ 30+34
+ 108
- 85
- 19
- 20
+ 256
- 329
- 847
- 395
-T 15S6
1
$E+ 70+74
- 922
+ 292
+ 975
+ 5348
- 1618
- 5728
-13320
+ 3632
+14602
'
^ ^-{-59-|-5J
%E~^-y -\-50-\-bJ
+ 172
- 74
- 133
- 594
+ 298
+ 402
+ 491
- 470
- 42
'
JE +90+94
|E ^+90+94
f+v''+90+94
- 541
+ 174
+ 564
+ 3856
- 1205
- 4055
-12092
+ 3608
+12887
'
if +30+24
+2041
- 431
-2290
- 5080
+ 499
+ 6274
+ 4928
- 1026
- 7597
rf
ii^"^? l~ w
~~ w~T"ty~T~ U
+ 384
- 384
- 1410
+ 1410
+ 2605
- 2605
*
5* V*i Q\~iA
- 131
+ 106
+ 69
+ 12
- 366
+ 350
+ 717
+ 772
- 1728
*
IE ^+50+44
+2169
- 596
-2295
- 8241
+ 1980
+ 9008
+12680
- 2086
-14823
rf
E ^+30+44
ff +^+30+44
- 389
+ 135
+ 383
+ 1251
- 479
- 1189
- 1212
+ 687
+ 930
*
ft +70+64
+1550
- 457
-1609
- 7940
+ 2211
+ 8376
+17170
- 4245
-18650
*
IE ( +50+64
- 349
+ 113
+ 352
+ 1665
- 543
- 1678
- 3127
+ 1052
+ 3117
rf
\t +90+84
\i (^+90+84
6 -)-^+90-(-84
+ 937
- 286
- 963
- 6044
+ 1784
+ 6274
+16950
- 4724
-17880
"'
i +0+4
?-^+ 0+ 4
+ 757
+ 514
-1583
- 3272
- 3644
+ 8214
r i
is +50+54
* i.'~|~oi7~i~oj
"^ ^~ }~ v i" OW~4~O J
+7767
-2522
-8820
-35692
+10085
+42033
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XVc Continued.
[Vol. XIV.
Unlt-l"
Sin
Uio
w
f
-i +35+34
is-0+30+34
-}+0+30+34
+ 4758
- 1369
- 6128
- 15945
+ 2307
+ 22836
1*
$ +39+34
$-0+30+34
is+0+30+34
- 882
+ 732
+ 177
- 280
- 2580
+ 3816
1J'
ft +70+74
jf-0+70+74
is+0+70+74
+ 7549
- 2504
- 8212
- 44427
+ 13864
+ 49192
?
-V + 8+ A
-i-0+ 0+ ^
-|+0+ + 4
- 32
+ 784
- 1031
+ 220
- 4194
+ 5051
f
$ +99+94
\s- 0+90+94
$+0+90+94
+ 5780
- 1929
- 6154
- 41583
+ 13412
+ 44735
v
i +0
$-0+ 9
-i+^+
- 768
- 1156
+ 1924
+ 2821
+ 6406
- 9227
ty
4 + 0+2J
|e-0+ 9+2J
. -l+^+ 0+2J
+ 209
- 771
+ 446
+ 1404
+ 3774
- 5879
,,'
J +50+4J
i -^+50+4J
-i+^+50+4J
-21869
+ 6125
+24564
+ 85960
- 19358
-101260
,,'
-J +30+2J
i-^+30+2J
-i +^+30+2J
-10182
+ 1576
+13528
+ 27638
+ 2276
- 43309
y
\t +30+24
4-t^+30+2J
j+^+30+2J
+ 384
- 939
+ 715
+ 3626
+ 3382
- 8550
V
f +30 +4 A
*-^+30+4J
i+0+30+44
+ 3250
- 1333
- 3256
- 10017
+ 4996
+ 9153
iV
f +70+6J
it- ^+70+64
j+^+70+6J
-23414
+ 7146
+25145
+122108
- 34410
-133985
*
-ft +
-J -V>+
-| + ^+
+ 768
- 2308
+ 1540
- 2821
+ 10637
- 7816
ili
| +90+84
^-^+90+8J
$+0+90+84
-18847
+ 5935
+19837
+122928
- 37138
-130949
i"
i +0+4
|-0+ 0+ 4
-l+^+ 0+ 4
+ 761
+ 906
- 1920
- 3333
- 5387
+ 9831
,"
is +50+34
$-^+50+34
-i+0+50+34
+15303
- 3577
-16828
- 49954
+ 7957
+ 58649
^ 3
-Jf +30+ 4
i-0+30+ 4
-f+0+30+ 4
+ 1582
+ 1300
- 3410
- 5765
- 6572
+ 14260
v a
\i -0+4
|-0- 0+ 4
is+0- 0+ 4
+ 451
+ 494
- 1096
- 1890
- 2861
+ 5381
mf
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
87
TABLE XVc Continued.
T,
Unit-l"
Sin
-
w
*.
1*
f +30+34
- 3918
+ 8760
is ^+30+34
+ 1588
- 5289
$+VH-30+34
+ 3637
- 6391
l"
ft +70+54
+ 18292
- 83098
|j-^+70+54
- 5104
+ 20825
it+^+70+54
- 19286
+ 89973
f
* +0+4
- 902
+ 3781
$!-</>+ 0+ 4
- 988
+ 5721
-i+#+ 0+ 4
+ 2191
- 10762
jl
J +50+44 - 1
+ 634
- 3482
fa ^-j-50+44 jf
+ 87
- 836
-j+^+50+44-.2f
- 933
+ 5479
f
-it +30+24-J
+ 428
+ 480
- 1816
- 2805
-5H-VH-30+24-1
- 1050
+ 5226
p
ft +30+34
- 1916
+ 8929
it ^+30+34
+ 2
+ 1220
it+^+30+34
+ 2553
- 13126
f
Jf +70+64 -S
+ 488
- 3307
I _^-|- 70+64 2"
27
+ 23
$+^+70+64 -2
- 623
+ 4387
p
-* +0 -J
- 475
+ 1965
55 ^+ JT
+ 1141
- 5536
-i+0+ -2"
- 508
+ 2916
p
it + 0+24+J
+ 1282
- 5447
$-<+ 0+24+J
- 90
384
| +^+ 0+24+J
- 1620
+ 7647
p
is +50+54
- 1544
+ 9111
^ ^+50+54
+ 222
- 735
f+^+50+54
+ 1838
- 11413
p
| t + 9g +8 j_j
+ 304
- 2460
Jt ^+90+84 J
- 42
+ 266
fj+^+90+84 J
- 364
+ 3013
,1
20+24
- 1955
+ 14862
60+64
- 35276
+ 189348
9
+ 3312
- 23724
df~\~A.Q ~j~4d
- 5097
- 4328
^+40+44
+ 6177
- 16310
C& ~f o V ~|~ 8 a
+ 45199
- 304998
iV
26+ 4
+ 6733
- 33547
20+34
- .3730
+ 1693
60+54
+142854
- 673242
<1> +4
- 9270
+ 61512
-# +4
+ 4207
- 28940
V^+40+34
+ 5323
+ 55061
-i+40+34
- 13730
+ 9080
V^+40+54
+ 22898
- 84425
V^+80+74
-200024
+1218446
7 ?"
20
- 3268
+ 14164
20+24
+ 3445
+ 15177
60+44
-190467
+ 772593
+ 12782
- 78712
^ +24
+ 5239
- 35125
^+40+24
+ 2712
- 60586
-Ji+40+24
+ 4409
+ 41693
^+40+44
- 52183
+ 143461
V-+80+64
+294332
-1600036
m'
88
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
TABLE XVc Continued.
Unit-l"
Sin
>o
w
r/ 3
26+ A
+ 3479
- 17883
T
65+34
+ 83314
- 283500
<!> + 4
- 7839
4- 47423
-<l>+46+ A
+ 6634
- 36904
V>+40+34
+ 27512
- 44330
0+80+54
-144023
+ 688658
fr,
25+24
+ 10709
- 50725
2(9+ A -2
- 1732
+ 8521
65+54 -.J 1
- 7799
+ 50227
^
- 12782
+ 78712
# + 4+JT
+ 11006
- 60629
4>+45+34-.T
+ 4022
- 29208
tV
-0+45+34 -2 1
- 3616
+ 27235
0+45+44
- 28408
+ 176052
0+85+74-2 1
+ 9526
- 75678
f if
28+ A
- 7475
+ 36068
26+24-2
+ 159
- 2967
65+44 -J
+ 11564
- 66719
+4
+ 11762
- 75153
^ +^
- 6024
+ 38182
-0+45+24 -.T
+ 7090
- 45771
0+45+34
+ 35006
- 199168
0+45+44 -I
+ 1108
- 281
i.
0+80+64-.?
- 15308
+ 111481
m'
An inspection of the preceding table, which is typical of all the trigonometric series under
consideration, shows readily that any function of this type is of the form
lie' sin K' + lJcsin (K</>) = 2k' sin K' + Ilcsin K cos d> +_ lie cos K sin </
or
IV cos K' + Ik cos (K<{>)=21c' cos K' + Ilc cos Jf cos ^Ilc sin if sin <j>
or, more briefly, a + 6 cos ^ + c sin ^
wherea, 6, care trigonometric series and can be written by inspection from the tabulated function.
Hence, in v. Zeipel's notation (Z 54, eq. 96),
T { = X { + Y { cos <l> + Z t sin <J>
and the integral may be written
W^ = x^+y^ cos ^+2< sin <[>
The functions T and W are to be used in this form in solving equations (95).
Considering only first order in the mass in T
T 2 = X 2 + Y 2 cos ([> + Z 2 sin ^
where
X 2 = Ilc' sin K'; Y 2 = Ik sin K; Z 2 = Ik cos K
or, X 2 is the part of T 2 which is independent of </>, Y 2 is a trigonometric sine series having the
same numerical coefficients as the part of T 2 which contains <[> in the argument, but in which
<f> is omitted from the argument, and Z 2 is the corresponding cosine series.
Considering the first two of the eqs. (95), the first one states that W 1 is not a function of
alone, or, W FW1 = n- W=rTF1
"l L iJ ", "1 L "iJ- t2+W
Making use of this fact in the second, Wj can be obtained from (95 2 ). (See Z 54.) Introducing
the auxiliary functions^ and u v defined by (99) and (101), the differential equation for TFj is
replaced by the equivalent differential equations, (100) and (102), for ip^ and u v .
The series
and
can be written by inspection from T 2 , or, better, the integration itself can be performed in part
at the same time.
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 89
The function fa is given by Z 59, eq. (103), or,
From the table of T 2 , page 82, it is not difficult to write immediately
The terms of higher order must be obtained by the usual method for the mechanical multi-
plication of series. A logarithmic multiplication is the most direct.
In each term in the expression for fa the terms of lowest rank must be of the first rank.
TfL *??? f ffL
Recalling the tabulation of factors in Z 53, w, , -^> -^> etc., are all of first rank. But the
coefficient for a given argument consists of three terms in ascending powers of w. Hence
fa w, within the limits of the given tabulation for T 2 , is of rank 1, 2, 3 for each order in the
mass. Table XVI, giving fa w, is tabulated with double headings. The three subheadings
indicate the expansion of the coefficients in a Taylor's series and the main headings give the
factors in the development of the radical in Z 59, eq. (103).
Having found fa, its reciprocal, fa- 1 , inclusive of first order in the mass, is given by
The second term is the negative of the first three columns of Table XVI multiplied by tff-*.
QAL
The product of 2 fa- 1 and that part of T t which contains <p gives -^, and integration with respect
to 6 gives u v tabulated in XVIII. The function u t is of first and higher rank because the factor
fa- 1 is of rank minus one and T 2 is of second rank.
From Table XVHI y l can be read by inspection, and ijy 1 added to Table XVI gives x lr
tabulated in Table XVII. The function W l is the sum of Tables XVII and XVIII.
In the integration those terms whose arguments are independent of are of the nature of
constants. In accordance with the condition that there may be secular terms in 0, the integral
contains such terms as the following:
'* _ _ 0-fcsin
As the constant of integration
-ifc sin
-
is added. Hence the integral contains terms such as
w
where i s the value of 6 for the time t = 0.
In passing, it should be noted that, in order that the expansion of Z 59, eq. (103), shall
represent the function, we must have
- t 4* V?
- i
- . u . W>
and this condition should be tested for a given planet before applying this method of determining
the perturbations.
To the computer the extent of auxiliary tables, the arrangement of series in logarithms or
natural numbers, in seconds of arc or radians, inclusive or exclusive of numerical factors, and
foresight in combining operations all these are of the greatest importance. But considerations
of this kind would carry the reader into complicated details which are best left to the com-
puter's own judgment.
On the other hand, general considerations about the extent of the published tables are of
importance in the discussion of the accuracy of the final tables. Yet, for a given limit of
accuracy, it is so difficult to determine, for each table, the highest powers of m', w, TJ, T)', and j 2
that little or nothing is said about it in connection with individual tables, but the discussion
is reserved until later.
90
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
TABLE XVI.
<t >l -w=x l -i 1 y l =[(l-e COB
Unit 4th decimal of a radian.
Prw
10-1
)-
to-*
vm
w
w
w*
w"
w
w'
w'
w
V*
-0. 0460
+0. 231
-0.52
^
-0. 0060
+0. 040
-0. 127
*r
4
+0. 0331
-0. 195
+0.53
ij
25+24
+ 42. 889
- 107. 72
+ 106.7
'
25+ 4
- 15. 427
+ 52. 39
- 75.2
i f
45+44
- 122.10
+ 484.1
- 813
-0. 0460
+0. 231
-0.52
9<
45+34
+ 357. 75
- 1183.5
+1650
+0. 0331
-0. 195
+0.53
*"
45+24
- 258.93
+ 687. 2
- 779
-0. 0060
+0. 040
-0. 127
;
45+34 -.T
- 14. 75
+ 71.7
- 164
l 1
25+24
+ 28.2
- 433
+0. 262
-1.70
+0.0003
-0. 0022
V
65+64
+ 428
- 2295
+0. 262
-1.70
+0. 0001
-0.0008
5V
25+ 4
- 316.1
+ 1592
-0. 767
+4.46
-0. 00021
+0. 0018
,y
25+34
+ 108.5
- 49
-0. 094
+0.69
-0. 00011
+0.0009
,y
65+54
-1889
+ 8902
1 t ,
-0.86
+5.2
-0. 0001
+0.001
-n"
25
+ 237.6
- 1030
+0. 555
-2.87
+0. 00004
-0.0002
,,"
25+24
- 125.3
- 552
+0. 276
-1.85
+0. 00008
-0.0009
,,"
65+44
+2770
-11237
+0.83
-4.7
,"
25+ 4
- 168.7
+ 867
-0.200
+1.21
,">
65+34
-1346
+ 4581
-0.200
+1.21
ft
25+24
- 389.4
+ 1846
-4498
ft
25+ 4-2 1
+ 126.0
- 620
+0. 032
-0.23
ft
65+54 -J
+ 113
- 731
+0. 032
-0.23
? rf
25+ 4
+ 362. 4
- 1749
p if
25+24 -JT
- 7.7
+ 144
-0. Oil
+0.09
}> ('
65+44 -.T
-i 187
+ 1078
-0. Oil
+0.1
m'
mf
m"
TABLE XVII.
Unit-l'
UJ I II}- 1
PrtO
I/OS
w*
w
UI
w'
w
U)
1)
25+24
+ 1179.6
- 2963
+ 2935
V
25+ 4
- 318. 2
+ 1081
- 1552
*
- 0.95
+ 4.8
I 3
45+44
- 3358
+ 13313
- 22356
- 1.27
+ 6.4
n'
4
+ 0.68
- 4.0
W
45+34
+ 8609
- 28481
+ 39702
+ 0.79
-4.7
^
- 0.12
+ 0.8
v
45+24
- 5341
+ 14175
- 16063
- 0.12
+ 0.8
?
45+34 -2
- 304
+ 1479
- 3383
t
25+24
+ 1955
- 14861
+ 7.2
- 46.6
?'
65+64
+11758
- 63112
+ 7.2
- 46.6
,y
25+ 4
- 6732
+ 33547
-15.2
+ 88.0
,y
25+34
+ 3730
- 1691
-3.8
+ 27.9
,y
65+54
-47616
+224423
-21.7
+130.
,,"
25
+ 3267
- 14165
+ 7.4
- 37.8
,,"
25+24
- 3446
- 15176
+ 7.8
- 52.5
??*
65+44
+63489
-257533
+19.0
-108. 1
ft
25+ 4-2 1
+ 1733
- 8522
+ 0.4
- 3.1
ft
65+5 J-2
+ 2599
- 16744
+ 0.7
- 5.2
ft
25+24
-10709
+ 50748
-123705
,"
25+ 4
- 3479
+ 17880
- 4.1
+ 24.9
I?
65+34
-27772
+ 94500
- 4.1
+ 24.9
? i
25+24 -JT
- 159
+ 2966
- 0.2
+ 1.9
f ri'
65+44-2'
- 3855
+ 22240
- 0.2
+ 1.9
? 1
25+ J
+ 7475
- 36070
(0-5 )sin
W'
A
- 570
+ 2421
4950
- 0.45
+ 2.7
-7.2
m'
m"
No. 3.]
MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
. ..,,. TABLE XVIII.
91
j=y, coe
An </>
Unit-l".
COB
M
m->
tr V
^
*
"
V
V
<J>+2d+2J
+ 294.89
- 839.5
+ 1229.8
- 740.6
+ 3328
- 4069
+ 734
- 5586
+ 5671
- 0. 316
+ 0. 114
+ L59
- 0.67
- 3.6
+ L8
r
-J+20+2J
+ 396
+ 978
+ 2940
+ 1494
- 7431
- 15782
- 9351
+ 23105
[+ 37112]
- 2.62
+ 4.42
+ 1.80
+ 16.8
- 28.4
- 1L7
li*
-++29+ J
V>+20+ A
+ 2068
+ 1492
- 2280
- 8658
- 10582
- 677
+ 11302
+ 40793
- 39010
- 13058
- 28348
- 83730
+ 6.18
- L91
- 5.57
-3.95
- 36.9
+ 13.6
+ 32.8
+ 23.6
e
<j>+26+2J
+60+4.1
- 1634
- 861
+ 6349
+ 7081
- 3794
- 25753
- 16199
+ 22127
+ 45318
- 4.04
+ 2.12
+ L90
+ 2L4
- 14.4
- 10.8
i
-4+20+ J-J
J+26+2J
- 866
+ 260
- 2677
+ 4260
- 1674
+ 12681
- 10988
+ 5101
- 30930
- 0.22
+ 0.07
+ L6
- 0.5
I
0+4J+4J
<p+46+4J
$5+80+8.1
+ 2549
- 3089
-LI 300
+ 2164
+ 8155
+ 76250
F-1L9J
'ill 1
+ 89
- 25
+ 70
i
rr
_^-f-40-)_3J
^+80+74
-11449
- 2661
+ 6865
+50005
+ 42212
- 27530
- 4540
-304611
+ L9
[+36.4]
-20.3
[+33.8]
- 23
-241]
+118]
-248]
if
^-|-40-|-4j
VH-40+2J
+26091
- 1356
- 2204
-73583
- 71730
+ 30293
- 20846
+400009
-10.1
f-25.51
1+28.01
1-4L9]
+ 83
+153
-153
+284
^
-fjfj^
-13756
- 3317
+36006
+ 22165
+ 18452
-172164
+10.1
-12.4
+16.6
- 65
+ 64
-104
i
111^
- 2011
+ 1808
- 2381
+14204
+ 14604
- 13617
+ 18919
- 88026
- L9
'it! 1
+ 5.7
+ 14
[- 14]
+ 10
[-42]
H
-#+40+2J-J
- 554
- 3545
+ 3827
-17503
+ 140
+ 22886
- 27870
+ 99584
+ 0.5
- L8
+ L3
[- 3.7]
- 4
+ 14
- 11
[+28]
,
,5 08U1
+ 767. 72
- 2820.9
+ 5210
+ L265
- 6.35
+14.3
V
v> + J
- 569.95
+ 2421.1
- 4950
- 0.455
+ 2.69
-7.2
t
+ 6624
- 47448
+23.8
[-22L 9]
?Y
t + J
[-18540]
+ 8414
[+123024]
- 57880
-73.4
+36.0
+572.4
-282.2
'
J +2J
+10478
+25564
- 70250
-157424
+55.2
+87.3
-374.8
-652.8
**
!> + J
-15678
+ 94846
-69.9
+438.6
/*!
t
+22012
-25564
-121258
+157424
+359162
[-511232]
+ 9.9
-23.1
- 77.0
+165.0
?
v +2
-12048
+23524
+ 76364
-150306
-251640
+498328
- 5.2
+14.8
+ 45.8
-112.0
m'
m' 2
92 MEMOIRS NATIONAL ACADEMY OF SCIENCES.
After the determination of W,, the function F 2 [FJ is obtained from the solution of
Z 53, eq. (95 2 ). The integral may be written as in Z 63, eqs. (105), (106), or, quite as simply,,
as follows:
F 2 ' = - a- cos e )(w+ F t ) -[(l -e cos
The function F 2 ' is given in Table XIX.
Anticipating some later developments, for which we shall need
[(l-e cos c)F]
the function
[(l-ecosOF,']
is tabulated in Table XX.
The determination of [ F 2 ] may be accomplished according to Z 65, eq. (108) Z 67, eq. (116),
or in the manner outlined below, which we regard as preferable.
Repeating Z 65, eq. (107),
in which all the known parts are contained on the right-hand side, the development of equivalent
equations proceeds in a manner analogous to that for TT,.
Writing
T 3 = X 3 + Y 3 cos <J> + Z 3 sin t}>
and introducing
and equating parts independent of 0, coefficients cf cos <j> and coefficients of sin </>, the three
equivalent equations are:
[(1 - J cos )(;+ FO^J^-t^J)^]- [(1 -
e cos
~ e cos
1 Vir- 1 - M^i
i ~3 - A ] 1+ 9 fll )Jw
[(1 -e cos e ) (w+ F^^J^-tZJ)^]- [(1 -e cos ) J ^[2
Multiplying the second of these by ij and subtracting from the first:
e cos
[(l-
[(1-6 cos )J(T 2 -[T 2 ])deJj+2[X 3 -r 1 Y 3 \-
MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 93
Multiplying the second by cos $, the third by sin tf> and adding:
A( tt _ WU)= _+(l-e C03 s)F l -r i F 1 + l + 2([yj cos ^ + [ZJ sin f)
F,)^ f { F, cos ^ + Z, sin v'--[F, cos
(1 -e cos )(+ ,) , cos ^ + , sn v'--, cos +, sn
in which
= [yj cos Hz sn
and [.XVI, [FJ, [Z 3 ] are read by inspection from T 3 , which is to be determined as follows:
If Z 50, eqs. (89), (90), are written in the form
_t i
-= [1 cos(/ w)] = 4U 2r cos e cos (s <!>)+.i) cos (2s ^)-f-ij cos t& +
cos 1 9> I
9/v*_ 1
1 n* 8 n cos s + 2 7/ 1 cos 2 2 cos ( A)
-8 if cos (e-^)+2ij cos (2-
then Tw and T' r , given by Z 49, eqs. (84), (85), in connection with Z 50, eq. (87), are given by
2V=-r,-4{l-2jj cose-cos (s-^)+ij cos (2e-v'')+7 cos <f>+ ____ }
IS P . ,(n + r.-n+s)iji^'/ 2 ' sin A
r = {3 + 14^-8^ cos s + 2jj cos 2s-? cos (j-^-Sij 1 cos (s-^)+2i9 cos (2s-^)+2ij cos ^
o
and r, (Table X\Tna) is computed by Z 53, eq. (94), in which
S
The function
is tabulated in Table XXI; the function
u = [/] cos
is tabulated in Table XXII.
From the latter [?/,] can be read by inspection, and ijfyj added to the former gives [xj.
Finally, (Table XXIIa),
F W t ] = [rj + [y J cos ^ + [zj shi ^
> * . : M M ^
L;.
94
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XVIIIa.
[Vol. XIV.
Unit=l"
Qin
ur*
r->
oin
w*
w
w'
to
w
Vfl
-+#
+0. 339
- 2.01
< +20+24
2t-v&+20+24
j+26+24
-0. 375
-0. 137
+0. 498
+ 2.403
+ 0. 847
- 3.223
+ 7.72
2i +40+44
t+^+40+44
-0. 438
+0. 429
+ 2. 234
- 2.338
2t+4>+60+64
+0. 361
- 2.372
20+24
f-H-20+24
- r+^+20+24
-0. 00047
+0. 0036
-0. 0123
+2. 199
+0. 286
-3. 294
-14.58
- 3.85
+23.82
+ 33. 34
t
#
-0. 00038
+0. 0035
-0. 0136
-2. 811
-0. 688
+12. 20
+ 1.67
- 16.51
!
e +40+44
0+40+44
-0. 00015
+0. 0013
-0.0048
+0. 432
-4. 536
- 2.58
+35. 80
- 95.79
3
e+0+20+24
+1.017
- 6. 333
9
t+^+60+64
-3. 219
+22. 43
\
25+ 4
I-0+20+ 4
- +0+20+ 4
+0. 00017
-0. 0014
+0. 0055
-2.520
-1. 253
+4. 372
+14. 78
+10. 20
-28. 56
- 31.95
*
t + 4
+4
+0. 00014
-0. 0014
+0.0060
-0. 404
+1. 188
+ 4.06
-11. 30
+ 34.07
<<
( +40+34
0+40+34
+0.00005
-0.0005
+0. 0021
-0. 224
+6. 480
+ 1.53
-47. 37
+120. 37
*
t+0+20+34
+0. 214
- 1.66
1
f+0+60+54
+5. 977
-36. 82
(0-0 ) cos
n
v
v
20+24
t-^+20+2J
- t+^+20+24
-0. 00188
+0. 0143
-0. 0489
-1. 141
+0. 235
+1.12
+ 7.14
- 1.12
- 7.39
- 20.54
v
<!>
-0. 00059
+0. 0051
-0. 0189
-0. 357
+ 2.62
-8.20
i
*
i +40+44
V&+40+44
+0. 00155
-0. 0141
+0. 0540
-0. 975
+0. 939
+ 7.39
- 7.27
+ 23. 43
>
*+^+20+24
-1.12
+ 7.39
I.
20+ 4
e-^+20+ 4
- +^+20+ 4
+0. 00068
-0. 0058
+0. 0222
+0. 847
-0.17
-0. 828
- 5.79
+ 0.93
+ 5.96
+ 18. 12
1
<l> +4
+0. 00021
-0. 0020
+0. 0085
+0. 265
- 2.10
+ 7.15
t
+40+34
^+40+34
-0. 00056
+0. 0056
-0. 0239
+0. 724
-0. 697
-5.90
+ 5.80
- 20.35
1
*+0+20+34
+0. 828
- 5.90
m' 3
m' 3
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
95
TAM.B XIX.
W.'
Unit-l".
e*
^
r
IP*
w
*
K*
*
"
-#+
+0. 2108
- 1.059
+2.379
^++40+44
-0. 2108
+ 1.059
-2. 379
y
l
+0.843
- 4.236
V
+40+44
-0.843
+ 4.236
iJ+ 20 24
- 294.9
+ 740.6
-733.9
-1.200
+ 7. 772
_^_j_ t -j- 20+24
-0. 875
+ 5.583
<^++ 20+24
+ 294.9
- 740.6
+733.9
+0.274
- L697
^++60+64
+1.800
-1L658
^+2*
-0.105
+ 0.529
+2+40+44
+0.105
- 0.529
Jjf
* + ^
-0.227
+ L344
if
e -j-40-f-3J
+0.227
- L344
if
^+ e 20 4
+1.758
-10.233
if
t!i+ +20+ 4
+1.083
-6.493
tf
^+ t+20+34
-0.204
+ 1. 377
rf
^+ +60+54
-2.637
+15.350
,2
-#+
+ 384
-1410
s
_^+ t 4^44
+1679
-6656
+20+24
+1180
-2963
if
+20+24
-1180
+2963
if
y''+
- 384
+1410
f
^+ +40+44
-1679
+6656
J
-#+ t - 4
- 285
+1210
-^+ -40-34
-2460
+8138
B jn'
+20+ 4
- 318
+1081
If*
-20- 4
+ 318
-1081
#+ + A
+ 285
-1210
*>!'
(f>+ +40+34
+2460
-8138
(0 ) sin
1
-tf+ -20-2J
+0.549
- 3.40
+0.00090
-0.0068
5
J+ +20+24
-0.549
+ i40
-0.00090
+0.0068
Tf
_^+ t_20- 4
-0.407
+ 2.75
-0.00032
+0.0027
Jf
#+ +20+34
+0.407
- 2.75
+0.00032
-0.0027
m'
m"
m
/j
96
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XX.
[Voi.xiv.
[(1-ecose) W 2 ]
Unit- 4th decimal of a radian.
0.
*.
-
-
-
w
w
wo
w
w'
,
w
+0. 01022
-0. 0513
+0. 115
9*
+ 18.6
- 68
+0. 187
-1.76
+0. 00043
-0. 0039
9 "
+0. 296
-2.46
+0. 00020
-0. 0020
-0. 186
+ 1.34
-4.8
^ Tj'
j
- 13.8
+ 59
-0. 529
+4.34
-0. 00075
+0. 0065
'20+24
- 14. 29
+ 35.9
- 36
-0. 1006
+0. 647
-1.91
-0. 000055
+0. 00041
^'
20+ 4
+0. 1377
-0. 811
+2.19
+0. 000020
-0. 00017
]j2
40+44
+ 81.4
- 323
+0. 477
-3.64
7 l'
40+34
- 119.2
+ 395
-1.295
+9.36
5 /J
40+24
+0. 921
-6.09
; 2
40+34 -I
+0. 036
-0.32
(0-0 )sin
,
20+24
-0. 0266
+0. 165
-0.49
-0. 000044
+0. 00033
n'
20+ 4
M.' 1 .il
!
+0.0198
-0. 134
+0.43
+0. 000016
-0. 00013
) ) 5
40+44
,..!,
+0. 151
-1.16
+0. 00031
-0. 0027
i) r;'
40+34
-0. 334
+2.47
-0. 00052
+0. 0045
9"
40+24
+0. 165
-1.24
+0. 00015
-0. 0014
m'
m' 2
m' 3
+
(V'.XW 'I
i of ;.
Ho.*.]
MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
97
*
2
I
8
o
kO i < OO
d MOO
7?
X
\
S
sr <M r-i
w .
>. 0i5
8 S OT
+ i +
E
-1
9
1
i
C4 O
o o
I-H CO IO
lOO
CH SO rl
l-l
+ 1
- 1
9
i-H r*- i-H <N 09 O 00 *O
CO CO C*J O ^ C^ <N N *& M 4 CO I s - Ol *T5 OS
OO< i O^^t-CCGO*O -^(NOC^C5O Ol^-t*
? ; [
s
OOOCOCOOOiOCio i-i-<o6t^OCi rH*-iO
+ 1+ ++ < 7++ i ++7 i + +1 +
g
c^ o "oi ifl r^ co^'i N re o ^r -^ i ^* N o co
Swc^ic-f^rcseow csoo^C'^'-o OO^H-^*
O O r-l O N 00 d ">" ^< S 1O CO 00 COC0O4
t
1
o o" o 1-1 i -<r * d o o N o o IN o o o
J^+ 1 + 1 +.1^ 1 +11 + 1 1 + 1
I
<MO"'m O -J-'a'co'T So"lM 00 "O V- C -
lOr^co^^rt^r- lOsco csoiodoo Nt^oo
OQt^O30-^r~i-H t^in^lOOt^ Ot^C<
O O -N id O> TT O OOMt^OCC OOO
'
* - ^
"Sog
O O O O O O i-l i-i<D O O O O O O OOO
+ i + ++ i ++ 1++.L -^ + i +
1
N 00 -^
O Mlfl
O C*5 ^
S o c
. 5 5
OO OO
+ 1 +
35s
1
3
WOOC: ^"t^CJCO CSOOt^ CC O5t^QO
2;oco o^*r~c<i <35 '^'^~'55^ "^* t ^2*
t
?
o'oo o' o o' o o o o' o' o o o' o o'
1 1 + 1 + + + + 1 1 +J_ 1 1 + 1
S
oo * N oo'ir'n^' t^ o a> ~O ^-'oc'o'
t^OO or^ccc^ COT^SSQ?'? . r^oo-^
III
o'o'o o' o o' o' o" o o o o o o o o'
++ ' . + -'.-l-l ' ++ ' + + 1 +
?
* s c
^^^^^,^4 i ^^^ i
N TCOC^CO CT? N 1M <I?
++++++ 1 ++++ + 1
^> ^ <d OS ^ ^ ^ ^> ^ ^ ^> ^ ^
ciCM^rr-^ 1 ^- c^iC^^^ -^r -^
a r^fr
x ^ v M C4 C
*=" B* *=* =- V S> V (B w V C>%> V f> V B- V B-
B* R B C- p- p. ^ fy. ff. ^ p.
'<^N
N
u
110379 22 7
MEMOIRS, NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
s
!
&&
?? /-
CO
-,
1 ~
CD
'
^-* " '
S
S
g
3
CD
rH
^ ;
8 .
:v'S
ivi? c," x r-t -!
*v C f-* C* C 1 "'
' ""*" ^ dr
1
1 +
W
1 " u ^'
'-*, O '" "^ X
-i- ( -J..4- i *.
f , -f r
3
O COCO
^ CO CO
CO CO W CO CM O
M 1 OS rH t>- CO i i
ooirio^ NN
?
i7-,'T'
j,ij;- >_ X
r- 'v tc v i, .T;
(-' -^t ;c i"* !'*
r.i - c- o o
+. 1 .+.
+ 1 1+ + I
I
-r. r i
COO Ci
rH CQ 1O
COITUS
IO rH C35 O ^ *O
rH OO TT CO U5 CO
O t^lN IO CON
S -J C*
'
*5f*' c^ >-- .- 1
T^t - } ta .- , <jf '\~,
C C .^ <** . ru
I Bill
V ^^ ^i.- V
'' + J-
rH O> CO O' O' O
+++ 1 1 +
tr ^
" *H-
^ C C 1 C 1 ':
- 14
9
1
S
IO CO CM
O * CO
OOO
CO
+4- 1
O CD O CO O CD
<f "5<M <N T)< (N
CNO NO oo
o' o o o o' o
I++ 1 1 -f
S
S
SSS
o'o'o'
1 1 +
O rH t^ CD N rH
o' o o' o o o'
+ 1 1+ + 1
1
c ~.
11 II1I
C- 0- C C
t. 4- -i ;
l|i
SS8 J < i
5ff
s
CM N CM
CN COCO
t-H OO 00 CO OO **
!> ift CO CO rH CO
^^ 1||1
J'j i? fj
r 5 if
o' o o'
OOOO OO
v w O O
H h 1
1 -f-f 1 1 +
~ z> o o C 1 o
=-oc
i i ! f-
f 4~r
1
s
7
o
^,^^, rj
o
E H
'"I* T" ^V
*^ CO U
!c' |
<BcTci J
a ""a 5
r" r' f
^t- < ^PC^'
tit -
+ + 4,
-f- -
4-+ ! -f4--t"H
S.-S.-3.-S- ^ -S.-S-
V
XV V
ff p* w
"'
~*
-a^-iT
R-
f=* P* K-
, *
f
iS
~ s
-SS
No. 3.]
MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
TABLE XXIIa.
99
P
^
>
.
-
#+20+24
- 0. 614
+ 4.059
-10.3
!
20+24
- 4.255
+ 2. 791
+27.89
-23.39
- 271.5
+ 167.4
+ 636.6
- 637.4
$
20+ 4
#+40+34
+ 5.444
-4.558
-31. 91
+33.80
'V
i K
' -3d) doiriw n
1
40+44
#+20+24
#+60+64
-#+20+24
+ 0. 11
+14.90
n bn <fvil>
L>a$i sv/ IF/
+1514
+1360
-1227
- 273
-5780
-3387
+6415
+ 179
'J
irf
4
40+34
#+20+ 4
#+20+34
#+60+54
-#+20+ 4
+ 0.13
-44.62
F
fj |, ^00 .,_ fl
I ni( l,;
291
+1974
- 222
+7160
+2452
+ 536
-9002
+1012
^*
7"
nr/7.x >I<JBT
40+24
- 0.06
+30.53
ro 4 I)j )>
,!* n.d.
* '(
40+34-2"
(0-0 ) sin
+ 0.34
^i^I + <lf)<
S 60') t- 1) ([
rj-W-w
. 6
|
20+24
#+40+44
- 1.64
+ L014
+ 0. 782
+10. 18
+ 8.43
-5.96
c.Ej-.irx
fiO^ J)T
<
^
20+ 4
# +4
#+40+34
40+44
+ L22
+ 3.249
- 0.579
'
+ 7.81
- 8.26
-30.12
+ 4.74
[(.'Wnr-.'WX
MO *-!)]-
[ !In dtldfi n
' L'W ] ,[,TT )
r V l.^ui/t
!$
4
40+34
+ 3.25
-16. 10
]-'^-V*i:
<*n*-nj-i
f,^ ^6 , -
)fiw-H
*
40+24
(0-0 )' cos
+ 7.64
if , 6 rs
i\t\ i "^^
,(,.-
1 , ;*lf 'Ms
eoo >- 1)4-
*{.>. lllftli-
#
d$n
- 0.356
+ 0. 266
+ 2. 62
-2.10
^ moil li^loq
qolavsb orfi
>J ijaliniifc nfi:
*nul firfT
niii n nl
Jaup') r.irfj 1<
m"
.1 i tuS
m'
In the construction of Tables XXI and XXII it is necessary to compute
:io1- L] uvti' --tj ."(.ijfttu'}/!l(.-> nf) T>J]
. f n-> noiJaup" '.JT
f(r 2 -
as one factor of a product, but the more complete tabulation is best arranged as follows. This
function gives all of the terms of the first order in the mass in W t -[ FJ. Let
100 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi.xiv.
and denote first order terms in F 3 -[F 3 ] and TP 4 -[FJ by W 3 " and W 4 ", respectively.
Then because of the similarity in the equations for these functions of successive ranks, the
sum
W,"+ W S "+W 4 "
can be computed by Z 70, eqs. (117), (118), (119). The coefficients F, G, B are tabulated in
Tables XXIII, XXIV, XXV. The mass factor ra' is, of course, implicitly contained in the
tables.
it
Eliminating the distinction between </> and , the function is
W t "+W 3 "+W<"
in which the coefficients A p _ g , determined by Z 71, eq. (121), are tabulated in Table XXVI.
The coefficients A M in the function
(l-cos) (F 2 " + W 3 " + W 4 ")
are computed by Z 71, eq. (123) and are tabulated in Table XXVII
By means of Table XXVII we readily compute
[(1-ecosO (F/'+F s "+W 4 ")]
tabulated in Table XXVIII.
Proceeding now to the determination of
[(1 - e cos e) W3
(from which we shall subtract [(1 e cos) W 3 "], already included in Table XXVIll), we have
by Z 53, eq. (95)
in which all quantities are known. The integration gives W 3 [ TFJ.
Having computed W 3 [ W 3 ], [ W 3 ] can be obtained from Z 53, eq. (95).
The function [T t ], computed from Z 53, eq. (94), is tabulated in Table XXVUIa.
In a manner similar to the development of equations for W, and f W t ], the right-hand side
of this equation, when computed, can be segregated into portions independent of tf>, terms
multiplied by cos </>, and terms multiplied by sin 0. It is of the form
A + B cos ^ + C sin ^
where A, B, C are too complicated to be written analytically, but can be written by inspection
after the computation has been performed.
The equation can then be written in the three following equivalent equations:
-
_ W ,A.
in which we define
NO. s.i MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 101
From the first two equations we compute
*i- *-*,-,(" (A-r)B)d0
Let J
3 = fc/J COS 4> + fcJ 8m #'
* cr T- Qt '
a
Then from the second and the third equations
cos $ + (7 sin - (^,
-
-si" !'
By inspection of , the function [yj can be written, and itfj/J added to [zj-ij[yj gives [zj.
Finally,
[W*] = [zJ + [yJ cos + [sJ sin ^
and
[(l-ecos)FJ
is readily computed from IT,, which is tabulated in Table XXVTII&.
But this function contains [(1 - e cos e) W 3 "], abeady included in Table XXVHI. By Z 69
Subtracting Table XXVUIc from [(l-ecos e) Wj we have
[(1-COS) (W.-W,")]
which is tabulated in Table
102
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
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104
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV
1
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K S S S S
No. 8.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
105
8:
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+ 1
C
*"
. .
g S
"?
g g g
. . . .
g
g g
I 4
rH rH
+ 1
1 1
+ '.7
+7+7
1 1 1
+ 1 + 1
1
+7
1 1
g
8 S
S $ S
g e^e.^
S S S
s s s
g
S S
g g
o o
**~^T~
**~^~G*~Q
z*~~~~~~
N N M
c c o o
o
o o
-
"
^
O
-i^
^W
-K-r
O O O
^
C O
en
IOJCBJ
z .n aoci,
>M
ir V N- I- t .-*
. J <v- :-.- r-r
-ii i v i-r C- -t
& c : o V
t:g
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY. LEVY.
TABLB XXVIII.
[(1 -e coe )( W t "+ W 3 "+ W t ")] Unit-4th decimal ot a radian.
Cos
W
w'
s
- 4. 1829
+ 12.406
- 16.58
5'
- 68.61
+ 347.9
V s
- 83.96
+ 413. 1
f
+ 83. 96
- 413.1
%9'
J
+134.0
- 714. 2
V
20+ 2J
20+ 4
+ 70. 842
- 42. 107
[- 165.57]
+ 147. 38
[+255.8]
-288.6
!V !
40+ 44
40+ 3J
-345. 88
+876. 64
f+ 843.0]
[-1481. 7]
i?
40+ 2J
-514. 54
+ 405.5
40+ 34 -.r
- 61.87
+ 273. 1
! f>! M i.
m'
,
TABLE XXVIIIa.
Unit- 4th decimal of a radian.
V
-t
i
I it T
Sin
w*
w
tc
w
A
^+20+24
-0
00005
+0. 00073
-0.0682
+0.4056
- 1 TT .01 4
| ^#3 .fi ' ! S1I-)
.0
9
20+2J
0+
-0. 3324
+2.1665
T!
^
+0. 3381
-2.5547
'
+L0220
-7. 370
?' ; SS+
20+ J
.0-
+0.2654
-1.846
i)'
^ +4
0+
-0. 3622
+2. 472
''
^+40+34
0-i-
-1. 2106
+8. 472
m
m
/7
109
0/00 .0-
;.f)0 I
110
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XXVIII6.
[Vol. XIV.
Unit- 4th decimal of a radian.
Cos
ur
w-i
w
U)
UI
>
w
tc
u>
w
- e+<f>
-0.00004
+0. 00038
-0. 0032
-0.0005
-0. 4803
t +20+24
2t-^+20+24
<j>+2d+2J
0.00000
+0.00001
+0.00004
-0. 00023
+0. 0237
+0.0050
+0. 0726
-0. 15101
-0. 0318
-0. 4507
+ 13. 16
- 0.81
- 30.86
+ 1.31
2s +49+44
t+4>+40+44
+0.00004
-0. 00038
+0. 0153
+0. 0181
-0. 0770
-0. 0814
+ 3.88
- 16.23
- 14. 38
+ 61. 80
2t+<fr+60+64
-0. 0088
+0. 0576
- 3.0
+ 15.7
7
9
IJ
y
20+24
t-4>+20+24
- t+<fr+20+24
1
.!! il ( / /
+0. 5242
+0. 1384
-0. 0508
+0. 0749
-3. 3539
-0. 7747
+0. 4660
-0. 2385
+ 13. 16
+ 14. 86
+ 58.25
- 30. 86
- 11. 30
- 170.9
5
5
. +40+44
^+40+44
+0. 1723
-0. 6378
-0. 8380
+4. 082
-154.6
- 16. 23
+ 589.2
+ 6L80
I?
t+^+20+24
-0. 1801
+1. 1267
- 7.71
- 6.66
3
e +^+60+6J
-0. 3275
+2. 032
+178.4
- 933.0
J
29+ 4
,-^+20+ 4
- +4>+20+ 4
r
-0. 6099
-0. 1412
+0. 1220
+3. 634
+0. 6843
-0. 8554
+ 10. 77
- 31. 34
- 49.04
+ 118. 9
I/
* + I
+0. 0524
-0. 4182
?
, +40+34
Vi+40+34
-0. 0411
+0. 7660
+0. 2314
-5.430
+221.0
- 694.3
9'
+(/>+20+34
+0.0460
-0. 3060
+ 14.12
- 26. 01
i'
f+^+60+5J
+0. 3718
-2. 0745
-287. 1
+1309. 3
(0-0 ) sin
_, _.
5
v
q
20+24
t-^+2(?+24
- t+^+20+2J
+0. 0490
+0. 0144
-0. 0542
-0. 2949
-0. 0705
+0.3582
5
t
+0. 5810
-4. 7017
5
5
t +40+44
^+40+44
-0. 0618
-0.0453
+0. 4650
+0. 3389
>?
+^+20+24
-0. 0010
+0. 0290
1
20+ 4
t-^+20+ 4
- t+^+20+ 4
-0. 0364
-0. 0107
+0. 0402
+0. 2398
+0. 0585
-0. 2889
*
^ +4
-0. 7670
+5. 4890
3:
t +40+34
^+40+34
.
+0. 0459
+0. 0336
-0. 3715
-0. 2709
^
t+^+20+34
+0.0007
-0. 0220
m' 3
m' 2
m'
No. 3.]
MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
TABLE XXVIIIc.
[(l-coet)Tr"]
111
Unit- 4th decimal of a radian.
Cos
10
Ml
va
V
20+2J
26+ J
+60.76
-20.57
-152. 6
+ 69.8
m'
TABLE XXIX.
T7nit-4th decimal of a radian.
Cos
te-
-!
w
*
w
<
1C
vfi
to
V
20+2J
20+ A
-0.00004
+0.00038
-0.0032
+0.5106
-0.6292
-0.0005
-3.0290
+3.463
+13.16
-30.9
(8-O t ) sin
r ~~ e '
V
20+2J
20+ J
+0. 0092
-0. 0069
-0. 0072
+0.0094
S MS 3 - 1 ~ -e J
" l/ *
"* "'":
m'
These developments cover the function 17 within the extent of our tables. This does not
mean that W is always inclusive of all these terms, but that these terms occur in one or more of
the tables. With the exception of [(1 e cos e) W], which contains W 3 W t ", W is to be under-
stood to mean F== Fj + W ^ + [ jpj + ( ^// + Wy + Wf ")
W= W,+ W t ' + [W3 + (W,"+ W t "+ F 4 ").
The ascending powers of w, TJ, 17', f are selected independently in each function.
To avoid along series which is analogous in construction to T 2 , the function W t " + W t " + TP 4 "
is not tabulated. The sum of this function and Tables XVII, XVHI, XIX, XXIIa gives W.
Since W is so long and we only need W, it is not tabulated. The function
W= TPj_,
is given in Table XXIXa.
It is convenient to collect here [(1 e cos e) W], which is required later. The function is
given by the sum of Tables XVI, XX, XXI, XXVIII, and XXIX, and is tabulated in Table
XXIX6.
We shall also need the function
Evidently S can be written by inspection if Wis tabulated. If the double headings are retained in
the construction of H the mass factors and ranks are explicit as in the construction of W. If W
is not given, we can write by inspection E, (previously required in the computation), E 2 ' and
[EJ from F u F 2 ', and [FJ, respectively. The remainder, namely, ,"+"+/', can
be written from W 2 " + W s " + W 4 ", i. e., by inspection of Tables XXIII, XXIV, XXV. The
function 5- E is given in Table XXIXc.
112
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
^
HI aojl
CO CO
CO CO CO
t".'
,..*,-. > -i..j
3
-f- 1 1
' Ia ! ** Ml l!i: "
i : I
OS OS OS
iO iC iO f i i O CO
OOO CMt^cor-co
CO CO CO CO CO
M
3
r-t ^ rH i I CO CM rH rH
CO" f-i 3* rH lO
pasn M3M, mnnjoo aiqj up sraiai aaxSap paooas XX
1 ++ + 1 1 1 1
I++++
sis s?? ss
U9t- S t~
CM CO CM CO I 1 -- iO CM CO
CM OJO WCO
f CO * 1C CO CM K5 00 O OOO O rH VC rH COCOOO OO CO CO CM
S
o o o" o* o* co o" t-t
oo" o' us' O CM'
O: o O' ^ CM O* rH rH 1C 1C' CO -^ t-^ l.O O* ^' rH ^' CO Q 1C CM
COCOrH rHrHt^COrHCM CO OrH
+11 1 ++++
+ 1 1 1 1
++I l+l 1 1 1 I+++I++I+ ++I 1
CO
00
i i
rH
i^ t~ CM O C*l OO OS t"- rH \fi CO CM CO CM GJ>
1
o*
O
O' rH* ^* rH CM* O* O' id rH CO CO O* O" CO rH
1 1 ++ 1 ++ 1 1 1 + 1 1 ++
3
CM 00
CM lO OS* bU3rHCOO
^ i t O CM ^* r-4 lO CO
CO ^f CM rH CD OS O
yoj<^. .0-r
1C O t^ CC IO
1 t *
CM CO
CO
1 1 + +++ 1 1
1 1 1 1 +
t--f-^;; ,
OS 10
S
id rH CO* lO rH CO OS CM
CM lO CO t 1 " OS CM ^ i-H
H pH ^ *
rH
?|S|S|
rH rH
CM rH O rH iO rH CO CO lO ^* rH rH CD CM *&
______)
T
++ 1 +111+
+ + + + 1
+ 1 ++ 1 1 + 1 1 + 1 1 1 + 1 1 + + 1 ++
rfT
S
i,J ilU'Hll
... ,
00
i<) i /i >ulv;
,pM (i y> 4-- I '] 1<> noiiqo'M.'i iii) J.ti// .
'da) oiii
<NCOt- rH
3
CD* CO CM CD* CO ^* O CM
OS rH CO CO t** rH, t~
CM CO OS r rH
r-* CO lO t^ rH
CM r-i O rH CM
co o co co en co co co co co oo CM rH oo t- oo rH a> CMCM^CO
lOCOOOCOOCCOrHCM CTSCMCMCTiCMGOSOCnCJlcN COCOCOO
r-COCMCO-'t 4 COCO CMi-HOrHOOCOt^t^CO t iOCOCM
OCO CM 1 ^ CMt^CMCMCOeft CMCM 1 ^ rHCOCMrH
rH CM rHrH CO rH T(<
lj iiu}:-.
hnjj
11+ 1 +++ 1
+ 111 +
1 + 1 1 ++ 1 + + 1 + 1 + 1 1 ++ 1 1 + 1 1
' ''/
OS
M
00 CO O
CM 00
S
Tp OS OS
CM rH 00
rH
+ + t 'r :i
2 S
CO <N
rH
l +
CO OS OS CO CD CM ^* CO O CO CO CO
"+++ +7+T+ 1 1 +
1 'J'i!l'".
,
wiT
i*f !>v.iu >-
I'v'-iv; : "
n 11 my\ ,- I)] ;mii )'J')Ho3 O-t ^lIOli!'* . ii- i
;'(')'/i^J r*t
(i h-
>J*;li
^X/ .\/7. ~A7. .Vf'A aoWuT 1<> >m;^
v'/l/.X
!
^ ^ ^ ^, ^ '^
^^^^^
^^,^^, ^,^ ^,^,^,^,^^^^, T ,^ I ^n-n
CM * O4 Tji CM CO
CO CO tO
TCMCOCM Tf 00 CO CO lO CO >O t~ CM CM -d
^ ^ ^i ^i ^ ^
ce> co co co
^ ^ ^ ^5 ^ ^S ^ ^ ^ ^ *^ ^) ^S ^S ^ ^S ^
+ ^. 4. + +
CM ^C^CO
TTCMCOCM -^00 -^C-JfMCOCM ^^J*OO -VCMCO
C,
" "(NCM
I'fXO ')
CM <M CM CM CM CM CM
1 1
jotjeui 'Jfl - 1 - ".";! nt.
B-
V
v
"s- i ,+>/>
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY
noiptujsuoo aq; nj
o >o eo i-i e> ! n o
+ +1 I + 1+ + I
3
v
i
oo
+ 1
M -
01 -f
k
H -f
^ ~-
o *~^ o ^^ *o C"4 25 10 o ^ ^ c
rt s-i o ^o oo c^ i < o 35 -^
< C^ <-^ ^ f^ fH
ii+7 i + i + i i +
Oil- CSJ 4-
~J
~
COWIM 00
I l+l
++
110379 22 8
'
^-i J
* f >
1
1-
113
114
MEMOIKS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
TABLE XXIXa Continued.
W.
fnit-l".
Cos
!0-l
ur-
w
w
w*
w<>
w
vfl
3
+40+44
- +40+44
+80+84
+ 2549
- 3089
-11300
+ 2164
+ 8155
+ 76250
-11.9
+ 3.9
-8.9
1
+ 70
(lOUOjJ/l ]
W
+40+54
+40+34
- +40+34
f+80+74
-11449
- 2661
+ 6865
+50005
+ 42212
- 27530
- 4540
-304611
+ 1.9
+36.4
-20.3
+83.8
- 23
-241
+118
-248
-M"
+40+44
+40+24
- +40+24
+80+64
+26091
- 1356
- 2204
-73583
- 71730
+ 30293
- 20846
+400009
-10.1
-25.5
+28.0
-41.9
+ 83
+153
-153
+284
1*
f+40+34
- +4(9+ A
+80+54
-13756
- 3317
+36006
+ 22165
+ 18452
-172164
+10.1
-12.4
+16.6
- 65
+ 64
-104
fr,
t+40+34-.
- +40+34 -JT
f +80+74-2"
+40+44
- 2011
+ 1808
- 2381
+14204
+ 14604
- 13617
+ 18919
- 88026
- 1.9
+ 1.9
- 1.1
+ 5.7
+ 14
- 14
+ 10
- 42
? -'
f +40+44 -.T
- c+48+24-S
+80+64 -.F
+40+34
- 554
- 3545
+ 3827
-17503
+ 140
+ 22886
- 27870
+ 99584
+ 0.5
- 1.8
+ 1.3
-3.7
- 4
+ 14
- 11
+ 28
(0-0 ) sin
- r-. -; o. g
n
20+24
+45+44
2f+20+24
+ 767. 7
- 2820. 9
+ 5210
+ 1.265
r * ? 2. 19
- 5.34
+ 0.78
- 0.55
+13.6
+22.7
- 6.0
+ 3.4
1
26+ A
+ ^
+40+34
2 +20+34
- 570.
+ 2421. 1
- 4950
- 0.455
+ 1.63
+ 5.94
- 0.58
+ 0.41
-11.0
-37.3
+ 4.8
-2.8
f
40+44
+20+24
- +20+24
2 +40+44
+ 10.93
- 2.19
- 1.92
+ 3.12
iV
?
4
40+34
E
- 570.
+ 6624
+ 2421. 1
- 47448
- 4950
- 0.455
+23.8
+ 5.94
- 23.00
-221. 9
- 7.2
,y
+ 4
- + 4
-18540
+ 8414
+ 123024
- 57880
-73.4
+36.0
+572. 4
282. 2
11"
+ 24
+25564
+10478
-157424
- 70250
+87.3
+55.2
-652. 8
-374. 8
,"
<+ 4
-15678
+ 94846
-69.9
+438.6
rt
'+ 4+^
-25564
+22012
+157424
-121258
-511232
+359162
-23.1
+ 9.9
+165.
- 77.0
f 1
+ 4
+ -T
+23524
-12048
-150306
+ 76364
+498328
-261640
+14.8
-5.2
-112.0
+ 45.8
(0-0 ) J coe
\>
t
+ 4
- 0. 356
+ 0. 26G
+ 2. 623
- 2.100
m'
m' 2
No. 8.]
MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
115
TABLB XX IXo Continued.
W.
as s
I 1
Unlt-1"
Cos
w*
W
w>
-
I
[+ 6+ J
- 293. 4
+ 913. 5
- 1400.1
.
t+30+3J
+ 338.1
- 2315
+ 9277
,
i+56+bJ
+ 42.9
- 284.3
+ 948.2
1+78+7J
+ 10.5
79.2
+ 288.5
5
$+30+34
+ 6172. 8
- 20580
+ 86549
-*+ 0+ A
+ 511. 2
- 2834
+ 7746
*+ 0+ ^
- 467. 9
+ 2335
- 6259
it+50+54
- 2217. 1
+ 23971
-157308
fc+30+34
5.8
+ 539
- 3713
S ^~ "r
|t+70+74
e!j- 364.3
+ 3259
- 15083
J
$+30+24
- 8375.5
+ 20591
- 95913
-fctJ
- 1023.4
+ 4443
- 10251
|+ 0+24
- 92.3
- 444
+ 3212
|+50+44
+ 3383.4
- 34097
+214736
|+30+44
- 138.6
+ 608
- 1089
$+70+64
+ 583.3
- 4805
+ 20748
f
i
<+ 0+ A
- 5022
+ 24269
l
+50+54
-31492
+154465
+30+34
+ 8169
- 18309
t+30+34
- 59
+ 7449
.
+70+74
+12392
-182737
+ 0+ 4
+ 1133
- 5174
|+90+94
+ 2342
- 25879
*?'
t+
+ 6153
- 26311
+ 0+24
+ 988
- 15732
+50+44
+88784
-357566
L5
_.
+30+24
-14498
- 3083
1 2 X
+30+24
- 1309
5
I& JK 'JC
+30+44
+ 4878
- 31947
t+70+64
-37540
+626187
:
+
- 3487
+ 12764
-1
f+90+84
- 7382
+ 77025
r
;
\t+ 0+ 4
- 5966
+ 27801
1
tf+50+34
-61877
+192684
\
t +30+ 4
- 1709
+ 26144
1 I
I
t- 0+ 4
+ 1693
- 6306
f+30+34
- 5297
+ 28649
$+70+54
+28418
-377278
f
+ 0+ 4
+ 6846
- 30542
+50+4J-J
- 3191
+ 15590
1
tf+30+24-JE 1
- 806
+ 10210
i
,+30+34
- 3829
+ 33852
;
,+70+64-2"
+ 932
- 14562
~'
,+ -2 1
+ 1762
- 6460
mf
X
116
MEMOIKS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
1
IO 00 IO O)
O
1
r.
CO ^f* CO CO rH
rH C6 00 INrH
8S
* '
rH CM rH rH
rH rH
-3
1 4- 1 + +
1 1
1
" O rH -"if
M-
rH kO
rH
L" ' \tf i-
C! r-J 01 OS CM
jrfo
COCO
COO*
*3
S
gq iS
CO OS
* CO
*? rH
&
4-4-4-4-1
1 4-
4-1
1 4-
1
IN
O IN CO O CO
CMOS
05 10
tO
rH CO
M
OS
CO
O
g
*4* CM Os O CO
CM co
COOS
g>0
coo
CO OS
rH rH
00
CM
rH IO
rH
TC CM
CM
rH CO
rH rH ^ rH rH rH
1 14-14-
4-1
1 4-
4-4-
1 1
4- 1 4-
1 1 1 4-4- 1 4- 1 14-4- 14-
+
O> T|-
08^ COCM
COIN
00 OS
BS
fl< OS
CO CM
CM
rH 10
COCO t~ O * 1* t~ CO S
a
O IO CO CM rH
^ rH CO
rH
*H
CM ^
t-ilO
rH
00 00
COCO OS
rH O OO
CO rH 00
t-lOO 00p CO OS CO IM t^ t- O O
COiMt- CO 1 '? (MOOr-t CO OO
(Mrtr rHCO rHCOrH CO rH
co
rH
CM rH
14-111
++
1 1
4-1
4-4-
1 4- 1
4-14- II 4-14-4-M 4- I
1
CO iO
rH CO rf
rH t- O CM OS
O CX ^ * CM
r-H CO
COrH
SS
rHO
gg
CO OS
Tf CO CM
t^-tO rHrHCOCO OSOS OOO
0000t~ CMCM IM CM OO rH O
1
a
O O O OrH
O CO
OCM
00
rH rH
^ O iO
CM rH ^< rHrH O O OO O O
o
4-14-4-4-
1 1
4-4-
1 +
1 1
4-4-4-
1 1 1 4-4- 1 1 4-4- 1 4-
+
-=>' 1C 1
x i*
R ?
j
J* rH t~
O O O O O
to co
^ CO
00
osco
co
(N IN
t- <<
CO ~ CO
t-ooo
IN CO
CO iO
IO CO O O C^ CM rH ^4 rH O
IO CM CO CM CM O O O O O O
S
CO
s
M
a
ooo oo
14-1 II
oo
oo
1 1
o
OO
ooo
1 1 1
OOO OO O O OO O O
4-4-4- 114-4- II 4-1
o
1
'
< c>
IO IN rH rH
IO CO
(M t^
CO
CO CO
1 *~ CO
s
* IM
IN 00
co os
Tj OS rH O
S S 88
88
S
8S
888
. . t ft*'
s
o o oo
oo
o o
o
oo
ooo
00 00
o
1 1
4- 1
."Ti
1 1
II 14-
+
tK-f-aJi- f
o
CO i-H
rH rH
CM rH rH
*?X> * rH tO
.
80
888
888 88
s
!*.'()!
nj.
;ww
o o
ooo
o' o' o' o" o
++
1 1 1
*
X
o
N
x, t*) ^^ a
1
II 1 1 '3
CM *
CO
<N
CO
IM CO
COlO
CM Tt* CO CM IO CM -^ cj^CM
5
-) (- -)-+ ++4- H h+ 1 + 4-
4-
IN <M Tf
TT
3
5
COCO
CM CO
coco cc.
(M IN CO
CMC-1CO CMCO CMCMCO (NCMCO -<N CM
co
<r
V
(=-
H
V V
B- "B-
p-
-
R '
*
S- K- K-
(S C
s*
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
117
>
d c^ o O t* t*
X
ic r^ oi .-i r-4 o
11+ + 1 4-
t
t-- to c^ o co
C5 cc ^ 00 t i ^
CC O O CO CD C**
X
O 00 * O O O
.
l-f 1 1 + 1
i
e o to S t-~ oc
1
C " < O O O O
11+ + 1 +
t"
e o 8 8 8 o
Ir
o o o o o
1 + 1 1 + 1
' 6
CO CO
OC CC t~ CC Tf
c'o" o o' o' o'
11+ + 1 +
C
S
s
+ + 7
^- i- *^s
v v v ?
B- P* F *=-
O (T- C-
.-.I
118
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XXIXc.
[Vol. XIV.
Unlt-1"
w-i
;-*
Cos
U)0
W
w<>
w
w
+20+24
- 90.5
+ 302. 7
- 478. 2
2t+40+44
- 26.6
+ 125. 5
- 270. 3
5
20+24
+ 589.8
- 1571.9
+ 1680
- 1.82
+12.00
1
+ 0.42
- 2.10
+40+44
+ 616
- 3638
+11175
- 0.42
+ 2.10
2+20+24
+ 23
- 161
+ 439
2 +69+64
+ 219
- 1451
+ 4616
*'
219+ 4
- 106.1
+ 360
- 517
+ 1.81
-10.64
t+ *
- 43
+ 161
- 269
- 0.08
+ 0.45
s+49+34
- 760
+ 3906
-10778
+ 0.08
0.45
2f+29+34
+ 52
- 143
+ 87
2 t +69+54
- 314
+ 1874
- 5403
f
-0. 317
+ 1.63
49+44
-1679
+ 7272
-13527
-0. 633
+10. 02
s+20+24
+ 274
- 63
- 1.20
+66+64
- 3474
+29267
+ 3.60
- +29+24
+ 1156
- 2171
- 2.40
2t
- 0.21
2f +49+44
+ 180
+ 113
+ 0.21
2e+89+84
- 1375
+11897
if
4
+0. 227
- 1.30
40+34
+3690
-12966
+19401
+0. 340
-19. 92
"" 5 ~
+20+ J
+ 222
- 1234
+ 1.96
+29+34
- 769
+ 2197
+ 0.01
% bS
e+69+54
+ 9240
-70866
- 7.25
- +29+ 4
- 646
+ 1806
+ 5.27
2s+ J
+ 99
- 444
+ 0.04
2t+40+34
- 109
- 922
- 0.04
2f+40+54
- 846
+ 4256
2J+80+74
+ 4012
-31827
l"
-0. 039
+ 0.24
40+24
-1780
+ 4725
- 5354
-0. 039
+10.42
s '3 ~
f+20+24
+ 499
- 649
- 0.32
t+60+44
- 5930
+40905
+ 2.86
-+20
-2.54
2+ 24
- 65
+ 285
2 t +40+44
+ 980
- 4150
2 t +89+64
- 2890
+20791
j 1
49+34 -2
- 101
+ 493
- 1128
+ 0.11
+29+24
+ 587
- 2983
t+69+54-J
- 193
+ 1759
+ 0.14
- +20+ J-2
- 0.14
2+ 4+2"
- 192
+ 705
2 t +40+44
+ 298
- 1876
2+80+74-J
- 65
+ 616
mf
m"
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY. LEVY.
119
TABLE XXIXc Continued.
fnit-l"
Cos tc
w
-
$ + 0+ 4
- 31.4
+ 131.
- 255
E+30+3J
- 48.0
+ 193. 6
- 360
"" J
+50+54
- 15.2
+ 81.7
- 201
[E+79+74
5.2
+ 34.7
- 107
1J
$+30+34
+ 1304
- 8173
+30282
-$+ 8+ 4
- 196
+ 506
- 598
$+ 0+ 4
+ 34
- 146
+ 292
$+50+54
+ 356
- 2212
+ 6781
$+30+34
+ 1
67
+ 294
$e+70+7J
+ 138
- 999
+ 3437
,/
(
^+30+24
- 1361
+ 7468
-25691
:
+
+ 29
12
- 155
_|-50.|-4J
- 482
+ 2635
- 7209
-i-39+44
+ 52
- 197
+ 280
$+70+64
- 207
+ 1348
- 4176
j>
$t+ 0+ 4
- 625
+ 3058
$+50+54
- 7151
+ 70387
-$+30+34
$+30+34
+ 5478
+ 18
- 5874
+ 1924
$+70+74
- 2111
+ 17665
-$+ 0+ 4
+ 187
- 590
f,e .
$+90+94
- 771
+ 6931
/
+ 0+24
- 231
- 1142
+17640
-159928
+30+24
- 9842
+ 1346
i
+30+44
- 892
+ 3699
e+30+24
+ 106
- 2494
j
j-j-70+64
+ 5918
- 45149
+
$+90+84
+ 2513
- 20914
^
^
k+ 0+ 4
- 507
+ 2729
1
+50+34
-10202
+ 84314
1
+30+ 4
+ 1055
- 678
j
- 0+ 4
- 100
+ 387
j
+30+34
+ 871
- 2817
+70+54
- 4065
+ 27951
f
$+ 0+ 4
+ 601
- 3122
i
+50+44
- 423
+ 4435
L QA J^O J V
f |~*X/-^fc d ~~~
+ 285
- 356
j
+30+34
+ 426
- 2410
.
+70+64-2"
- 108
+ 988
~ n
*+ -2
- 106
+ 402
m'
tnlqii
fn;
120
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
TABLE XXIXc Continued.
i-
Unlt-l'
Cos
..
w -
*
w
^
,
f
20+24
+ 1568
- 8912
+3.6
-23.3
60+64
+ 5879
- 31559
+3.6
-23.3
gY
20+ 4
- 2385
+ 11662
-4.7
+26.6
20+34
+ 2238
- 1015
-2.6
+18.4
60+54
-21644
+102003
-9.9
+59.1
1) I)' 3
20
-0.2
+ 1.7
20+24
- 1723
- 7588
+4.0
-27.1
60+44
+25396
-103013
+7.6
-43.2
V s
20+ 4
- 1160
+ 5960
-1.4
+ 8.3
60+34
- 9257
+ 31500
-1.4
+ 8.3
} ijj
60+54 -S
+ 1040
- 6697
+0.3
- 2.1
20+24
- 5354
+ 25370
-61855
sew
j 1 tf
20+ 4
+ 2492
- 12023
20+24 -2
- 53
+ 989
-0.1
+ 0.6
60+44 -J
- 1285
+ 7413
-0.1
+ 0.6
1 j j t>
L" *-h
(0-0 ) sin
tui
4
t -ft
! Writ
i)
20+24
- 0.55
+3.40
swi
1
20+ 4
- i <!.'.':
+ 0.41
-2.74
jf
40+44
4 (-.".
- 3.12
+20+24
- 1.10
-+20+24
- 1.10
Tj tf
4
- 569.95
+ 2421. 1
- 4950
+0.45
+ 5.94
40+34
- 5.75
+20+ 4
i ~ii~
+ 0.20
+20+34
'''*('
+ 0.82
-+20+ 4
t* ^- $
+*|
+ 1.01
,/J
40+24
'i(l!
t +e
U-: -j ft
-i i
+ 2.55
+20+24
soot
I.""' ^ ft
' 1 -f
- 0.15
-+20
- 0.15
Si'U;
(0-0 ) a coa
i'.'it'
"i U~t *
U*
<*f.<;
i"^82
i. tt fti
1 4 -
>) "?'
J
di'f"
U' '. *
- 0.26
HK*'
801
'** - L3 --ft
'"
':(
+ 0.20
TO'
m' 2
COMPARISON OF TABLES.
As a computer would discover in constructing tables, and as will be evident from an appli-
cation of the method to a planet, the coefficients in Table II and others of the same form are
given with unnecessary accuracy. Although so many digits would never be required, except
in a much more exhaustive development, they are given, for completeness, as they resulted
from computation.
In all the tables whose constructions involve the multiplication of trigonometric series, the
errors are difficult or impossible to determine. Although v. Zeipel's manuscript, which the
author generously furnished for comparison, is of assistance, the computations are not entirely
parallel, and comparison is not always possible. Many of the computations are so long and
NO. 3.] MINOR PLANETS LEUSCHNER, CLANCY, LEVY. 121
complicated that the origin of certain discrepancies is obscure. Aside from possible errors of
calculation, differences are due to the independent adoption of the highest powers of m', w, ij, 17', f,
and the number of arguments in a given series or product of series. In most cases our series
are more complete than v. Zeipel's. Whether or not the extension of the tables increases the
accuracy of the result remains to be seen from future applications of the theory.
Tables II-XV. -The discrepancies seem to be due to v. Zeipel's errors of calculation and to
their subsequent effects. The larger number of these errors have been traced in the manuscript.
Tables XVI, XVII check satisfactorily.
Table XVIII. The bracketed quantities in the first three columns are in error through
previous discrepancies. We did not discover the source of the general disagreement in terms
of the third degree, second order in the mass. These terms do not affect v. Zeipel's subsequent
tables, since they are of order higher than have been included.
Tables XIX, XX agree satisfactorily.
Table XXI. The discrepancies are numerous and their origin is obscure because of the
very long computation involved. In addition to performing a complete duplicate computation,
the terms of first degree and a part of the computation of second degree terms were checked
by the solution of the differential equation in the form given in Z 64. With the exception of
three or four single instances, the discrepancies occur in two groups, having the following
probable explanations. The neglect of the term
in Z 65, eq. (109), by v. Zeipel accounts for one group of differences. The other group can be
fairly well explained by an error in the addition of second order terms in +- fa to #, -^#..
A &
Assuming that for these terms he added w<t>, and, correcting his table, three discrepancies are
removed and two others are improved.
Table XXII. Considering the disagreements in Table XXI, Table XXII checks satis-
factorily.
Table XXIII-XXVn. These tables, like II-XV, are simple in construction, and the
discrepancies are due to errors of calculation, or they are the result of previous ones, with the
exception that some quantities have different numerical values because they are more inclusive.
The latter have been indicated by ( ).
Table XXVLLi. The discrepancies arise from the quantities in parentheses in Table XXVEI.
The omission of the term depending upon the inclination is justifiable in view of its magnitude.
Table XXIX. The discrepancies are numerous and striking, but, since v. Zeipel does not give
the formulae of computation, they remain unexplained. The remark is made (Z 77), "Die
Berechnung der Funktion [(1 e cos ) ( W 3 W 3 ")], welche eine sehr komplicirte war, wird hier
nicht im Einzelnen mitgetheilt." For this reason the development of the formulae which we
used has been included and the auxiliary functions 2[TJ, W 3 , [(le cos e) W s "] have been
tabulated. The differences are not serious because of the high rank of the function. Our
table is deficient in certain terms whose computation would be long and the omission of which
is justifiable in view of their magnitude.
',i-ffttl -jilj i: i :;// i <: r\ v;?"t nvtA'fi
PERTURBATIONS OF THE MEAN ANOMALY.
For clearness some of v. Zeipel's developments will be amplified and repeated in an order
which we found more convenient.
The determination of the disturbed mean anomaly is accomplished with the integration
of Z 9, eq. (47), (which implicitly contains Z 8, eq. (38)). By Z 9, eq. (43),
d = %(e-esms)-g' = 1 s g-g'
The differential equation is repeated in Z 78, eq. (124), in which is emphasized the fact that
d W
the arguments are functions of both e and 0, as is the case for r-
122 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi.xiv.
If we observe the character of as it is expressed in the definition and recall that we have
admitted trigonometric terms in 0, multiplied by t, it is evident that this argument, which is
a function of the disturbed positions of the planet and Jupiter, is not periodic, but varies con-
tinuously with the time. In the foregoing equation g and g' can not be regarded as angles
which are always less than 360. contains, therefore, a nontrigonometric secular part in e
and a periodic part in 6 and s.
If we write
6=(0-[0]) + [S]
[6] contains the secular term in s as well as periodic terms. The segregation of terms of
different type can be made explicit by the introduction of
fe Z 78 ' !
where i? is a function of s and d lt 2 , 6 3 are the periodic parts of ff [0], i. e., they are
entirely trigonometric functions of e. This covers the condition that 9 t can not include trigo-
nometric secular terms in e. By definition of tf and Q i
i
*? = [fi = [/??,)] - ^^
ds [_dJ ds
where [n'tis*] is the long period term between Jupiter and Saturn.
The derivative of (125) is
KJ n,. <-u<>;-4 ; .fe'nnoionib ro quuig aiio aininmA laquhx .7 vu ,(60Ij ,p<j ,t .. :v IK
,
^ = ^ + ^^
or ds odds
<pT' ; ibe, be, bo, \ / be, be be, \d&
= I -^ * -J 2 4- Tr-5 4- 14-ll-J ' -4 A 4- \-r
\bs be bs / V d$ 5$ d?? /ds
Expanding F(6,s), eq. (124) in a Taylor's series in ascending powers of 0<and making the above
substitution for-y- (124) becomes (126), in which
hi) liiiv/ ,-ino v:j.vv--nq lo J(;j ';'". '>il) -JTB T->I!> in .noifjiliJ >?> lo snon-j at emh <nu 8?j! r >aBtjOTO8tL
j Q
From the Taylor's series T- is written m (127). This is the differential equation for tf, the
a
right-hand side of which can be computed.
Substituting 3- in (126) and equating functions of equal rank, we have the differential
' TOD gJttlHil
equations (128j 128 3 ) for 0<, which can be integrated in succession.
Before integration we convert eqs. (128) into differential equations for ndz as follows:
ii'' r*Siou ifiiii ori.) lo i , .. , . ,. . . , Joo WIR
n8z = (7^3 - [n<52]) + [72^2]
' r . , 00 /, , .N
= 7n?2j+ r^2 2 + n^2 3 H ------ \-[n8z] Z 88, eq. (144),
where n8z t is not only a function of first and higher orders in m', in which the lowest rank is i,
but is entirely trigonometric or periodic. Then
r iri* : 'f" o
2 r
Z 9, eq.(46) gives flfe-[nte]-r |^ x (*>) + 0, (&,e) +6 3 (&,e) + ...... +wi? sin e+ (n'8z' -[n'dz'])
and
[n52] = r f^{t>-| + [7i'fe']4-c'-//c] Z 88, eq. (145),
where it is to be noticed that [ndz], unlike [ W], is not free from terms in e. Subdividing the first
of these two equations according to rank, we have Z 79, eqs. (130), in which n'dz' + [n'dz'] can
be neglected.
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 123
Differentiating eqs. (130) partially with respect to e, substituting in eqs. (128), evaluating
the right-hand sides of eqs. (128), we have eqs. (131, 131,), in which the superscript indicates
that only terms of first order in the mass are included, and where the argument tf replaces
the argument 8.
For purposes of calculation, the integrations are arranged as follows:
In
+ W 3 "+ F 4 ")
consider first only W t "+ W 3 " + W t " in the integration of eqs. (131). The integrations will
concern only part of the terms of first order in ndz l + nJiz 2 +ndz t . It is shown by v. Zeipel that
the integration for all three ranks can be performed conveniently at the same tune. Let this
part of the function be indicated by enclosing it in ( ). The integral
+
which is a trigonometric series, is given by Z 80, eq. (135), in which the coefficients L p . q are
defined by (136) and are easily derived from Table XXVII. The coefficients L p ^ are tabulated
in Table XXX.
The remaining terms of rank one which are of first order only, namely, ndz^ (ndz^), are
given by the first of Z 81, eqs. (137), in which TT,, IF,, [FJ, can be written by inspection
from Tables XVH, XVm, XIX, XXIIa, The function is tabulated hi Table XXXI.
The remaining terms of first order in ndz 2 and ndz 3 are given by the sum of Z 82, eqs. (139)
and (140). The function
___
is given in Table XXXH.
These developments complete ndz (1) within the limits of the tables, and we next consider
ndz (2) . We shall limit ourselves to functions in which the lowest rank is the first or second.
Consequently, ndz^ contributes nothing.
m 't
Anv function of second order in the mass and first rank must contain the factor r and in
itr
the given F (t>, e) this factor occurs only in Wf. We have, therefore, by Z 80, eq. (131,), for
one part of ndz^\ indicated by parentheses,
>) = f{(l -e cose) F^-tU -ecoas) W]}dt
This function is tabulated in Table XXXIII.
124
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV..
co-
co
i* -T
* .0
CO
2
o
o co"
rH
CN 2
O 1 CO CO CS CM lO CO CO O CM O **< lO CO lOt* <M OS
CM CO CS CM CD wS ^* rH COCNlO
+
+ 1
1 +
1 + 1 ++ 1 + 1 + ++ 11 1 1 + + 1
g
CO CO
OS OS
t^
o
' CO
CO CO
O CO r- 1 COOOCS CSrHOO ^eNt^-O CD t*rH iCW ^COCO COCOlCt^-
(>JOO COCNJlO lOO O3 rH iTiCOCO QOrHCOlOrHt--.
O>
OS rH^ i-HCO CO M C^ICS lOCOCO
rH N rH b
+
+ 1
1 +
1+ I++I +1+ ++I.I 1 1+ +i 1+7 +11 +
i
CO lO
1O CO
CO
Ij
O CO
CO i-i
OOJO -^C^^iO <MCOCN C'-COC^ICTS CN OOrH HH > 1 C^^d O CO CO i 1
GO
CO
COf-H COCOC^ rHCOTt< lOfHrH rH O IOO3 C^rH ^COlCCO
^ CNO MrH t^ lO COCO *OCOlO
Ci CO rH CO N t*
CO CO
+
+ 1
1 +
1+ I++I +I+++II 1 1++I l + l +11 +
CO OS
rH CM
CD
CM
O OS
rH lO
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r-l
rHO *^COC^ iiO-CO rHlJSc^^T 1 C^ lO COM CM rH COOOrH
r-l
rH
iO -^CO COCO rH lO O rH t^- t^Ot
CO I s - C^ itrH^COMCO
^* cs
+
+ 1
1 +
i + i ++ i + i + ++ i i i i + + i +7+ii +
rH
00
CO C<>
OS CO
CO
to
CO
O CO
OS
CO CO
^s
II1TJU5 rHt^-OiO fH-^CO O rH O rH -^ ^i CM COCO iQOOO CO'JJ-^cO
tN.^'cOlO^CO MrHCM OSrH frHCO
M CD ^ rH
^
i
1 +
1 +
1 + 1 ++II +1+++II 1 +++I +1++1+ +
rH
U5 t-
10 10
00
"5
rH
rH
CO O
OJCOO CO^rHCN I>- CO 1C COCNrH-^ O* COM COCO COCO^ rHCOrHO
CD US -^rHO OCS lO CMrH t-rHO CO 1 ^ COOO4
OOrHOarHOO COCOCOCNIOCMCD
1O rH rH rH C
lO C^J
+
1 +
1 +
1 + 1 ++ 1 + + 1 + + 1 1 i 1 ++ +1 + ! + t 1 + 1
CO
CO GO
TJ* CO
CO
rH
CO iO
CM CO
s s
CNCM rHiacp 1 -* 1 COOJCO lOCMt^-OS Oi 2^ U^rH CSCO I>-COOCN
f
rH
rH
15
t^- rH C^ s " CM rH CO
rH ^<
+
1 +
1 +
i i ++I++M+III i+i+i i+ i ^+7
S3
iO CO
*< CO
in
8
CO CO
r-i CO
CO CS
00 00
OS
COrHCM t^t-' COrH* rH OlO CO r-ICO t^CO COOSCC IO(MCO~
1OOS (N*t*CO 1OCO1O CN lOC^l CO WCM rHO COOSCM COrHCM
rH
rH
rH
T)CO i-irH COCO rHrHO * CO-9- COt~
IM t- rH rH CO *-'
rH
+
1 +
1 1
+ 11 +1+ +11 + 1+ 1 +1 ++ 1 + 1 ++ 1
i
w
m cxi
rH __ _
S
r-i r-i
SrHl-*- CO CO t 1 -^ COCO COrHrHiO t^ CO t CO COCOi 1 CMCMCDO
CO
rH
rH OS CO rH O
i
+
1 1
+ 1 + 1 + + + ++ ++ 1 + + 1 l+l +J^ +11 +
1
tO 00
CO 00
ss
lO
00
rH
OOOlCJ CO OSt^- OSOCO CMCOrH CO ^O CO CMOSCO CM OrH
lOCOrH CO lOOS CMCOrH rHCOO rH COCO CO t-COlO b* rHCM
CM i ( ^ CO lOt^- rHCTS-^ lO^TCM ^ O rH t* TfCOrH O rH^
f (
O CMrHrHrHCO rH CMrHCO rHCOrHfiM^CO
rH rH kA rH
i
+ 1
1
+11 1 +1 + 1 + +++ + ++ + 1 ++ + ^+
id id
C~J CXI
CO CO
cIS
CO CO rHCOOOrH CC CO r^t^OO JA^ OO CXI (M OSCNfMCS
o
o o
rH rH
l>t~ COCO COCO <Nr-t-<M
CO CO
+ 1
1 +
+111 ++ 1 ++I+I I++I 1 + ++ 1 1
O C
rH ft t t rH rHrHrHrH rHrHrHrH
^ ^ ++ II ^^+11+ ^^ ^.^ ++ 1 1
e e
r-l rH
g g gggg(M (M gggg ggrHrH g g gggg
*
_^_^
1 1
+ 1
se
^-.'ff-j* .... ggg .... g .. gg -g- ....
1
+ 7
I 1
+ '. i +1 + 1 L '. '. +1 + 1 '. +7 L '. + L i +1 + 1
s
s s
~
gj?S ggg JSSSS ?S SS SS SSS SSSff
o
*~o *~o
*~~^ *~~-i
^"o *^o ^"o ^- s ^- ^^ **~Z- ^"tN s *w > *^ ^~o ^^o ^^e **"o ^~o ^~c ^*o *~~^- ^~^- ^~o ^o ^~o s ~i x ~-C ^^ **~
l-f
MM
iJW
-~~ ^~ ^-~ - ~- -- ^~, jt-3 -^ j^ !*-C i-C ;-*; Ik^ <; -C -^ -C ^ ^ -; -c ^ *-:;---
(fl JO^DB^
H
Vo. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
125
17.3TX ajaT
00
rH
{'''/***;-
i-. eo 3?
I-*- CO CO
CM t * O
rH
m
OJ CO iS Ol
rH rH,
-.
1 + 1
1 1 + +
+
41 1 +
"" iV* 4 tV--*!;
1^5 i** i&
O 01 00
S3S2
_- c^> oo
- .\ll
t i ( N
CO
.
1 47
1 1 44-
+
+ 1 1 +
sss
rH rH rH
Bo^0
I
&* ro O5 iO
" CO d rH
CO t-
CM
00 O
_
H
-. r
i + i
1 1 44
+
II 1 +
-..-
"
OS t-
*? fl "*"
rH t2 2
^H i ^ s
r^ C 1 ! CO 'i'
^
^
S S
1 + 1
i i ++
4
II 1 +
t -4- ?
WU*I -i- V-i-j
00 iC
CC Cl
OO * CO T
r t^- ifi- 10
sr 3 o
CO O) '
>a
"& *&
OS
eo co
- *_ , - Hi*
1 + 1
1 44 +
+
1 J- ' +
1
Sig
seS
1
CM t- 2 s5
^H)
D
r^
*|So5
H
" S* 00
JS
rH
rH
a
08
1 44
1 + + +
4
1 + $
^>
.3 r.000fi--
t-t CS -n*
O CM CO
CC 5S r^
tO t^ t~~
t- OS rH
CO lO
1
XX OO rH
C CM CO
r- CO -V
cp C5
55 O
s
"
1
1 + +
1 + 1
4
1 4 II
|
.3
SI
O CC
CO OO CO O
t^. t*- ^^ Ou
M O 5^ -H
rH kO CC
rH i-^ CO
1
P CM *
C7S CO t~
**** CM O
5S
1
9
1 1
1 1 + 1
1
+ + 1
fjsti; Z tv-t-f-'
?,2g
<N -H OO
v * <y
*
"" Z 3
9
CC 1O O
co r- CM
>o
~. ~ ^
"3
a
1 4 1
1 1 1
1
1 1 1
B
a ;.,;
S
rH rH
liii
^* ^* rH rH
4 1
~ x,
i + 1 +
+ 1 1 +
i
C3
iZZ
S.
4-774
.2
CM* sr
+ -, 1
g s s e
1 1 1 1
"i"? r7r7
II +1
3
a
s s e
1?
8 8
S
I I 1
s e e"
rH T-^ ^^ ^H
4141
s e s s
1
8
rH rH . .
+1 '. '.
s e s e
r
1
i o o
I I S I
|
CO
CI
^ ^ -^
** ** ^ ^
'^
i*-i ."*^ N 1 ? [^
^=
c_,
r" JaptU
126
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XXXI.
[Vol. XIV
Unit-l"
Sin
*-.
.
w
w*
+2,5+24
+ 294.89
740.6
+ 734
B
+4,5+44
- 839. 5
+ 3495
- 6224
2+2<5+24
- 147. 4
+ 517
- 737
ft
e +4
+4,5+34
+ 1229.8
- 4069
+ 5671
j) 3
- +20+24
+ 784
(- 3570)
(+ 10522)
+2,5+24
+6,5+64
2t+40+44
- 202
+ 2940
+ 415
(- 1657)
(- 17009)
(- 2587)
(+ 13183)
[(+ 43527)]
(+ 6440)
2f
- 192
+ 705
i) i)'
- +2*+ 4
- 2386
(+ 11567)
(+ 37527)
+2,5+34
+ 1492
(- 968)
(- 12562)
+20+ 4
+6,5+54
- 1962
- 8658
(+ 9257)
(+ 42767)
(- 23263)
(- 92732)
2+40+34
- 615
(+ 3264)
(- 6905)
2 +4
+ 142
r- 605
B
- +2,5
+ 1634
- 7081
+ 16199
+20+24
- 861
- 3794
+ 22127
+60+44
+ 6349
- 25753
+ 45318
f
- +20+ 4-2
+ 866
- 4260
+ 10988
+6,5+54-2
+ 260
- 1674
+ 5101
+20+24
- 2677
+ 12681
- 30930
B
+40+44
+ 5907
- 11149
- +4,5+44
- 269
+ 5158
+8<5+8j
-11300
+ 76249
i 5 S
B B
+40+54
-11449
+ 42212
+40+34
-11270
+ 951
_ +4,5+34
+ 1744
- 23941
+80+74
+50005
-304611
ij ij' 2
+40+44
+26091
- 71730
+4,j+24
+ 3985
+ 16118
_ +4,5+24
- 3137
+ 35021
+80+64
-73583
+400009
ij"
4-4^^-3J
-13756
+ 22165
- +4^+ 4
+ 3317
- 18452
+8,5+54
+36006
-172164
?i)
+4^+34-2
- 1707
+ 13125
+4^+342
- 2112
+ 15096
+8,5 +74 2
- 2381
+ 18919
+4^+44
+14204
- 88026
j 1 T)'
+4,5+44-2
- 554
+ 140
+4^+242
+ 3545
- 22885
+80+64-2
+ 3827
- 27870
+40+34
-17503
+ 99584
+(<5-i5 ) C06
1?
- 767. 7
+ 2821
5210
1}'
+ 4
+ 570.
2421
+ 4950
1*
2<
- 384
+ 1410
- 2605
n i
2+ 4
+ 285
- 1211
+ 2475
rf
- 6624
+ 47448
Y
+ 4
[+17970]
[-120603]
+ 4
+ 8984
- 60301
i?"
+24
-10478
+ 70250
f
-25564
+157424
>/*
+ 4
+15678
- 94846
,7*5
+ 4 +1
-22012
+121258
-359162
e
+25565
-157424
[+511232]
/" ^'
+.T
+12048
- 76364
+251640
+ 4
-23524
+150306
-498328
m'
NO. s.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 127
TABLE XXXII.
>o->'.-rt-{<' e; r*^
Unlt-l".
r
oox
f'r%
!f
Sin
it*
.
_j-2i>+2J
' - ' 294. 9
+ 1036
" *" *
,
+ 384
- 1410
*.j-wiT
nAfusi }.ij (
2+2tf+2J ' '''
+ 1679
74
- 10348
+ 149
'! ma
isftil'i fiiijJu
T .vti'io i^iTt i-i
/""
t + ^
- 285
+ 1211
- 2460
+ 13067
((!> "' t\'j HfX
-' ' ' ,^
?
+2i>+2J
- 101
- 883
+2<>+2J
- 978
+ 6459
fife ' p'J KK)
+60+64
- 8820
+ 424
(+ 77487)
- 1332
2f
[+ 96]
I- 352]
ff'
/ (1* t '
- 2068
+ 8418
-i-2i>-|-3J
- 1492
+ 2460
fe.Q_l_R 1
c ~r "" i J
+ 2280
+ 25974
12618
(-206223)
i to ic
r* baa ooil
9 jf
- 615
t- 7 1]
+ 1420
[+ 303]
1*
- +2*
+ 1634
- 5447
+2tf+2J
+ 861
+ 2933
<? ' jW frt>
j-6fl-|-4j
- 19047
[+134400]
/
+2J+ 4 Z
+ 866
- 3394
(j . * . *
^_6tf+5J S
- 780
+ 7362
j-j-2<>+24
+ 2677
- 15358
^
+4tf+4j
- 5098
+4i>+44
+ 4499
^il y ' ~
+8+8J
+ 45200
r ,
?Y
+ 40 + 5j
+ 22898
v/
-j-4t)^-3J
+ 5322
'' ; '1 I ' ,=.
e+4t>+3J
- 11270
f -j-8iJ+7J
-200020
lr."W.
1 f 7 *
+4t5+4J
- 52182
+4i?+2^
+ 2712
^jiixiu
+4^+2^
+ 4408
+8i9+64
+294332
V
+4i>+3J
+ 27512
I Y. 7 7. .7.7
7. wI<!
_ _i-4^-i_ J
+ 6634
+W+54
-144024
w
+4t>+3^-2
+ 4022
r . BaM
+4J+34 2
- 3616
.<! ni f-.
+8t?+7*l 2
+ 9524
+4d+4^
- 28408
. r
y* f 7
e +4$+4j 'Z
+ 1108
+4i>+2J 2
+ 7090
+8i>+6J 2
- 15308
+40+34
+ 35006
m'
ori.i
run-
-HT
The coefficients in parentheses differ from v. Zeipel's values because they contain additional
terms. See p. 134.
128 MEMOIRS NATIONAL ACADEMY OF SCIENCES.
The remaining terms in the differential equation for ndz^ are, by eq. (143),
(1 - e cos )
-(!- cos .) tF,'< 2 > +TW - F/') - =T /" Tf t < + i (
all the terms of which are of the second order whose lowest rank is the second. They therefore
contain the factor ?
w*
To obtain ndz^ it is necessary to return to eqs. (124)-(130) and make developments for
terms of the second order similar to those for first order. The resulting differential equation is:
~n&, = (1 ~ C 2 COS) {7^<') - (nte/O) } Wf > - (1 - e cos ) w
{
roi
*vr
' - 1 - e cos
-[(l -e cos e) ^
The sum of the last two equations, when integrated, gives the terms of second order having
m' 2
the factor ^ It has been shown by v. Zeipel through computation and we have shown ana-
lytically that
and
[(1 -e cos s) F/ 1 ^!^^ 1 )- (nfe t ('))} +w^(n5 2 / 2 )) =0-
Therefore,
= l-e cos
-[(!- cos
The integral is tabulated in Table XXXIV.
Summarizing, we have included first order terms in
given by tables XXX, XXXI, XXXII and second order terms in
given by Tables XXXIII and XXXIV. The addition of Tables XXX-XXXTV gives the short
period terms in nfe, or, the function
ndz-[ndz]
which is tabulated in Table XXXV.
Returning now to the differential equation for tf, the evaluation of F (#, e) and its derivatives
in Z 78, eq. (127) gives Z 91, eq. (146). The variable does not appear; -j is a function of t? alone
Therefore the function is of long period. The integration is one step in the determination of
[ndz], the long period terms in the perturbations of the mean anomaly.
The function [(1 -e cos e) W] is tabulated in Table XXIX6.
The function f(l - e cos i)( W- ^sV W+ ^S\], computed from Tables XXIXa and XXIXc,
is given in Table XXXVI.
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
129
The function (1 - e cos e) (0, + 0, + 0,)
First, 0-
is computed as follows:
5W
is given by Z 93, eq. (150) by means of Table XXXV, and -=r= is readily written by inspection
of Table XXIXa. Performing the indicated multiplications and retaining only the terms which
are independent of e, we have the required function as tabulated in Table XXXVII.
By eq. (146), the sum of Tables XXIX6, XXXVI, and XXXVII, multiplied by the factor
gives <?(?), tabulated in Table XXXVIII.
TABLE XXXIII.
Unit-l"
Sin
tc-
te
U*
u>>
I?
f+4<+4J
- 0. 316
+ 1.59
-3.6
*
e+4tf+3J
+ 0. 114
- 0.67
+ 1.8
f
- t+2iJ+2J
t+2t)+2J
+6t>+6J
25+40 +4J
+ 2.62
+ 4.42
+ 1.80
+ 0.16
- 16.8
- 28.4
- 11.7
- 0.8
+ 1.8
*v
- +2t>+ J
+2tJ+3J
t+2^+ J
+6>+5J
2s+4^+3J
- 6.18
- 1.90
- 5.57
- 3.95
- 0.06
+ 36.9
+ 13.6
+ 32.8
+ 23.6
+ 0.3
- 0.9
1"
- e+20
J+20+2J
+6<>+44
+ 4.04
+ 2.12
+ 1.90
- 21.4
- 14.4
- 10.8
f
- +20+ J-S
+60+5J-S
+ 0.22
+ 0.07
- 1.6
- 0.5
+ (0-l> ) COS
5
- L265
+ 6.35
-14.3
l'
+ ^
+ 0.455
- 2.69
+ 7.2
V
2
+ 0.63
- 3.2
+ 7.2
v
2+ A
- 0.23
+ 1.3
- 3.6
1'
f
-23.8
+222
v
+ 4
- + J
+72.9
+36.5
-569
-285
v
+ 2J
-55.2
-87.3
+375
+653
*"
+ J
+69.9
-439
ft
+ ^+^
-9.9
+23.1
+ 77
-166
;* -!'
f +^
+ ^
+ 5.2
-14.8
- 45
+112
m"
110379 22 9
130
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XXXIV.
[Vol. XIV.
I noii-i
Unit-1"
Sin
.-
UJO
M
.
1" j-.i FiOJ ci;,i!(: t >-, !<
2;+4t?+4J
- 0.614
- 0.079
+ 4.06
+ 0.40
-10.3
'
E+40+4J
- 0.74
+ 1.74
+ 0.31
+ 0.45
+ 3.7
-18.1
- 2.0
- 2.9
*
j-i-4,+3J
2 +6i?+5J
+ 0.30
- 4.26
- 0.66
- 1.8
+32.0
+ 3.8
'
- +20+24
2+80+8J
-6.4
+ 6.4
+ 5.1
- 1.4
- 2.2
rV
P t -;-
- +20 + 4
+2<>+ 4
+6.5 +5J
2+8tf+7J
+12.0
- 0.9
- 8.5
-11.8
+ 3.4
- 0.8
+ 6.5
f
+ 2J
^-2t)+2J
+6iJ+44
- 5.1
+ 1.3
+ 6.4
*
- +20+ J-S
s-{~6t?~l~5^ S
2f -(-4^ -|-4d
- 0.3
+ 0. 3 '
+ 1.4
+(tf-l> ) COB
j
'
2+2i?+2J
- 1.02
- 0.78
+ 0.41
- 8.4
+ 6.0
- 2.5
''
+ 4
2t+2tf+3J
-3.25
+ 0.58
- 0.31
+30.1
- 4.8
+ 2.1
'*
_ +2i?+2J
2
2 +4<>+4J
+ 3.6
+ 1.1
+ 0.5
- 0.8
:'_'
"'
- +20+ A
2+ A
- 3.4
+ 1.6
L f .
,
t
- 0.36
+ 2.6
''
+ 4
+ 0.27
- 2.1
1
*
was.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE XXXV.
Logarithmic. niz-[ntz]
131
Unlt-1".
Sin
KT
r
vr*
V
w>
f
-+ *
4.1570
4.8741,
+ tf + 4
2.7684,
3.3827
3.7172,
1?
+ >+ J
4.0056,
4.7686
V
+ + 4
4.0766,
4.8295
f
i + + ^
4.1365
4.8738,
n*
+ tf + 2J
3.3345
4.5162,
rf
E +30+24
4.2240,
4.9611
5.6685,
1
+3i>+3J
4.0671
4.8483,
5.5636
5"
1 J+W+3J
5.0926,
6.0018
^
< +W+4J
5. 2325
6. 1714,
V
1 +5i>+54
4.7675,
5.7344
J 1
.E + W+4J-J
3.8050,
4.7998
^
-Jf+ 1>
3.3112
3.8350,
4.1355
?
-i-e+ J+ J
3.2065,
3. 7910
4.0833,
*
-,+30+ J
3.5338
4.6236,
*v
-< +30+24
4.0879
5.0382
^
+30+34
a 6012,
4.5318,
j 7
-J +30+2J-J
3.2074
4.1925,
^
1
9.868,
0.5689
2.922
3.4600,
3.3670
^
+ ^
9.482
0.2533,
2.673,
3.2959
3.1772,
it*
+20+ J
0.746,
[1.384]
3. 2927,
[4. 14906]
[4. 6990,]
+20+24
9.788,
2.47560
3. 10847,
3.4540
[3. 3960,]
>?'
+20+24
0.645
U-342,]
2.305,
[3. 6179,]
[4. 4018]
v 1
+20+24
0.326
1. 119,
2.935,
3. 3017,
[4. 39206]
j*
+20+24
3.4276,
4.23764
4.76933,
:r*
+20+34
+40+24
0.28,
1.102
3.1738
3.6004
[3. 5449,]
4. 27485
[3.8446,]
A
+40+34
+40+34
9.057
0.692,
3. 10161
4. 0519,
3.9302,
3.7975
4.52415
[4. 78162,]
<
+40+34
4.1385 n
4.6961
>* v
+40+34
4. 2431,
5.1290
ij
+40+44
9.500,
0.522
2.9351,
3.8035
4. 41616,
4.63017
q 3
+40+44
3. 7714
4.2108,
^**
+40+44
4.4165
5.0931,
A
+40+44
4.1524
5.0661,
iV
+40+54
4.0588,
4.8136
j*l +40+34 -J
3.2322,
4.2342
; ^' +40+44 -.T
2.744,
[3. 0%2]
5" +60+44
0.28
0.64 n ]
3.8027
4.77998,
5. 52852]
i) ij' +60+54
?* +60+64
j E + 60 + 54-.T
0.596,
0.255
8.8
1.070]
0-8 n ]
9-3,]
3.9374,
3.4684
2.415
[4.94342]
[4. 50125,]
3.48a,T
5. 70347,]
5. 27451]
4. 2931]
5" +80+54
4.5564
5.4999,
i) i? 77 +80+64
4.8668,
5.8416
T,V +80 + 74
4.6990
5.7030,
if +80+84
4.0531,
5.0844
j 2 ij' +8i>+6J-^
3.5829
4.6352,
fr,
t+80+74-2 1
3.3768,
4.4540
> ' 1 ", '-,
V-
$
V
- +20
- +20+ 4
- +20+24
0.606
0.791.
0.418
fl.422,]
1. 690]
[1.365,]
3.2132
3. 3777,
2.894
3.6657,
3.8866
[3. 4616,]
19260
4.72168
3.8078
p
- + 20+ J-J
9.34
0.28 J
2.938
3. 4714,
3.7862
1*
- +40+ 4
3.5208
4.07255
<
- +40+24
3.4965,
4.59582
-/v
- +40+34
3. 2416
4.5467,
s 3
- +40+44
2.430,
3.9848
j 1 ?'
- f +40+24 -I
3.5496
4.19852,
/*!
- +40+34 -I
3. 3247,
4.05994
132
Logarithmic.
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XXXV Continued,
niz [noz]
[Vol. XIV.
Unlt-l"
Sin
w -
^
^
u-
w
,
*e+3t?+2J
3. 6731
4. 0029 n
f-)-3''-i-3^
2. 3528
3. 2475 B
3.9005
7;'
$+30+3J
3. 6181 n
4. 2122
J 3
fs+Stf-j-SJ
J
3. 4072 B
4.4000
1) y'
^+3d+4J
3. 5244
4. 4012 B
TI'
fe+5i>+4J
3. 3533
4. 4231 n
5. 2725
rj
|-i-5t-t-5J
3. 1780 n
4. 2730
5. 1359 B
Tj' 3
fs-j-7*-(-5J
4. 2775
5. 4708 B
IT
i-j-7i>+6^
4. 4051 B
5. 6177
fj+7<?+7J
3. 9296
5. 1605 B
,
2e+2tf+2J
9.486
2. 1744 B
2.708
[2. 889 n ]
2. 599 n
rf
2j-(-2i>+34
1. 946 n
2.501
2. 516 B
M!
2IK+4J
8.8 B
[0. 561]
8.90 B
2. 789 B
9.599
[3. 5813]
1.711
[4. 1074 B ]
2. 5795 B
3. 1726
T
2+4,+4J
9.2
[0. 34 n ]
9. 819 n
2.618
0. 5840
[3. 4962 B ]
2. 7821
[4. 0890]
4. 51865
1
2f+6t+6J
9.653
0. 4645 B
2. 5979 n
3' 6265
4. 38424 B
| -(-5^+5J
1.2340
2. 1166 B
2. 7076
i)'
4-|-7#+6J
2. 3679
3. 3518 B
4. 0587
>)
|+7t?+7J
2. 1758 B
3. 1926
3.9204,
(l>-l> ) COS
,
E
0. 1021n
0.728
2. 8978 B
3.4504
3. 7168 B
I 3
e
1. 377 B
[2. 346]
3. 8211
4. 6762
jj 7) /a
5
1.941 B
2.815
4. 4076 B
5. 1971
frj
e
1.364
2. 220 B
4.4076
5. 1971 B
5. 7086
rf
+ J
9.658
0. 774 B
2. 7836
3. 3840 n
3. 6946
7) 3 7)'
+ J
1.863
2. 755 B
4.2546
5. 0814 B
^ /3
t+ ^
1.844
2.642 n
4. 1953
4. 9770 B
j" 5'
+ ^
1. 170 B
2.049
4. 3715 n
5. 1770
5. 6975 B
I$i
+ 2J
L742,
2. -574
4. 0203 B
4.8466
/ >)'
*+ ^
0.716
1.65 B
4.0809
4. 8829 n
5.4008
fl
1.00 B
1.89
4. 3427 B
5. 0837
5. 5553 B
,V
-t+ ^
1.562
2. 455 B
3.9535
4. 7803 B
?v
2
2.+ J
9.801
9.357 B
[0. 43 n ]
[0.473]
2.5842
2. 4548 n
3. 1493 B
3. 0830
3. 4158
^'^ e - 1. :|
9.56,
0.42
V
!+ j
9.43
0.32 n
1 sn A.Tg.+(#-d )2w*TiPi)'<lj 1 tC. 1 coa
where C lt C 3 , C 3 , represent the respective coefficients.
3 sin Arg.
Mil
No. 3.)
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE XXXVI.
-![(! -t coe ) (W- J- 2 ) (W+J- H)]
133
Unit 4th decimal of a radian.
Cos
K-<
tc->
to-
to- 1
w>
w
-
+0. 000032
-0.0080
+0. 0493
- 0. 176
+0.52
7, J
-0. 00028
+0. 0037
-0. 133
+L10
- 8.8
V
-0.00014
+0. 0026
-0. 095
+1.27
-14.4
? '
-0.0003
+0. 139
-1.20
+ 5.9
iV
J
+0. 00047
-0. 0070
+0. 252
-2.51
+22.8
/ V
n
20+24
+0. 000017
-0.00042
+0. 0437
-0. 366
+ 2.10
'v
20+ A
-0.000006
+0.00045
-0. 0639
+0.508
- 2.79
/
40+44
+0.00004
+0.0006
-0.194
+1.64
-11.4
V
40+34
-0. 00012
-0. 0012
+0. 372
-3.59
+32.2
*>
40+24
+0. 00011
+0.0003
-0. 252
+2.40
-19.8
f
40+34-2
+0.00001
-0.0001
+0. 032
-0.19
+(0-0 ) sin
*V
4
-0.00004
+0.010
- 0.08
Ji
20+24
+0. 000066
-0.00060
+0. 0399
-0. 275
+ 0.94
v
20+ A
-0. 000024
+0. 00047
-0. 0296
+0. 221
- 0.81
/
40+44
-0. 00023
+0. 0028
-0. 114
+1.02
-4.7
iV
40+34
+0. 00039
-0. 0053
+0. 251
-2.20
+ 9.9
!>
40+24
-0. 00011
+0. 0024
-0. 124
+1.11
- 5.1
(0-0 ) a coe
I- t..
7)'
-0. 00017
+0. 0014
-0. 052
+0.38
- 1.4
|V
A
+0. 00019
-0. 0021
+0. 077
-0.61
+ 2.4
*"
-0.00005
+0.0008
-0. 029
+0.24
- 1.0
m
m' 3
m", m' 2
m r '
TO"
m
TABLE XXXVII.
[ (00) (I ecoee)-gj-
Unit 4th decimal of a radian.
Cos
.,
-,
to-
~
J
w
,
+0. 000042
-0. 01071
+ 0.0883
- 0. 402
+ 1.31
- 3.9
9 1
-0. 00043
+0. 0056
-0. 189
+ 2.73
- 51.3
+ 299
1 ! /2
-0. 00021
+0. 0048
-0. 296
+ 4.47
- 59.8
+ 416
?
-0.0004
+0. 186
-2.00
+ 11.7
- 40
Tl 71
J
+0. 00076
-0. 0110
+0. 530
- 7.59
+104.2
- 682
Tl
20+24
+0. 000055
-0. 00086
+0. 1005
- 1.153
+ 21. 86
- 81.5
+217
If'
20+ 4
-0. 000020
+0.00090
-0. 1377
+ 1.463
-9.50
+ 44.2
-176
Tj 3
40+44
-0.00031
+0. 0041
-0. 477
+ 6.49
-133. 8
+ 708
J) 1)'
. 40+34
+0. 00068
-0. 0084
+1. 295
-17.43
+261. 3
-1266
"
40+24
-0. 00030
+0. 0041
-0. 921
+11. 58
- 95.2
+ 452
'
40+34 -S
-0.00001
+0.0001
-0. 036
+ 0.58
-5.3
+ 25
TJ 7}
4
0.00000
-0.0004
+0. 052
- 0.44
+ 2.0
TI
20+24
+0. 000044
-0. 00052
+0. 0266
- 0.212
+ 0.83
Jj'
20+ 4
-0. 000016
+0. 00038
-0. 0197
+ 0. 170
- 0.70
r 3
40+44
-0. 00031
+0. 0037
-0. 153
+ 1.62
-7.9
1j Jj'
40+34
+0. 00052
-0. 0072
+0. 335
- 3.40
+ 16.9
- .7.77,
!'!
1*
40+24
-0. 00015
+0. 0032
-0. 165
+ 1.68
-8.6
m' 2
TO' 3
m", m' 2
TO' 2
m' 2 , m'
m'*, TO'
m' 2 , TO'
134
Logarithmic.
MEMOIES NATIONAL ACADEMY OF SCIENCES. [VOLXIV.
TABLE XXXVIII.
0(0) Unit- 1 radian.
Cos
w-
te-
tO-4
w-
U)-l
w-i
w
w
w
1.5
[3. 909]
4.960
6. 6748 B
7. 2764
7. 540 B
7.31
'V
2.0
1.9
[4. 644 n ]
3.41]
[5. 160]
4. 75 n
6. 150]
6. 509]
8. 048,,]
8. 2077 B ]
8.838
[8. 994]
8. 655 B
8. 919 B
8. 100 B
?
^ .M -
2.83 n
5.146
6. 299 n ]
7. 994]
8. 740
[8 r 656]
1*
J
2.34
[4. 446]
[4.57]
6. 728 B ]
8. 4022]
9. 1999 B
9.0854
8.079
1)1'*
20
1.6
[2. 6 n ]
5.744
6. 535 n
8. 3811
9. 1031n
9. 0128
,'
20+ 4
0.8 n J
[3. 068.]
[5. 2988]
7. 2212 B
[7. 3772]
[8. 0372]
[8. 764 B ]
8.668
*
20+ 4
2.32 B
3.30
5. 886 n
6.718
8. 5059 ra
9. 2804
9. 201 7 B
V"
20+ 4
5. 301 n
6.149
8, 2302 B
9.0154
8. 938 B
P ('
20+ 4
8. 5592
9. 3245 B
9. 2428
f"
20+24
2.48
3.40
5.422
6. 292 n
7.476
8. 664 n
8.636
1)
20+24
1.22
[2. 94]
[5. 1206 n ]
7. 6416
[7. 9638 B ]
[7. 083 B ]
[8. 645]
8. 582 B
,,"
20+24
1.9
[3. B ]
5.442
6. 328 n
8. 0915 B
8. 630 B
8.742
ft
20+24
8. 5904 n
9. 3489
9. 8024 n
9. 6532
,Y
20+34
2.04 n
3.00
4.98 n
5.89
8. 0326
8. 1973 B
7.69
X
20+ 4-2 1
4.51
5.42 n
8. 1011
8. 873 B
8.792
? *'
20+24 -2 1
4.04
5.00
6.89 B
8.182
8. 158 B
?
/
40+24
40+34
40+44
40+34 -2 1
[2. 66 n ]
[2. 72]
[2.20]
1. 5 n
2.7]
4. 369]
4. 624]
2.45
6. 1031
6. 2526 n
[5. 824]
4.68
[8. 4188]
8. 5594
[8. 0924 B ]
7. 1747 B
[8. 5297]
8. 7988 n ]
[8. 4338]
7. 301
6.0]
7. 94 B ]
7.24]
8.111
7.90 n
8.287
7.74 B
8. 127.
8.210
8.044,
V 3
60+34
5. 30^
6.149
9. 1294 B
9. 7728
9. 6609 n
?r"
60+44
5.92
6.74 n
9. 4432
0. 14644,,
0. 05077
,v
60+54
2. O n
3.0
5.93 n
6.79
9. 2774 n
0. 03298
9. 9494 B
?
60+64
2.0
3.0 n
5.420
6. 292 n
8.634
9. 4351 B
9. 3608
?V
60+44 -
4.04 B
5.00
8. 272 B
9. 1028
9. 0334 B
h
60+54-JT
4.51
5.42 n
8.0554
8. 926 B
8.864
(0-0 ) sin
*v
4
[2. 60 B ]
4.71
5.94 n
6.507 B
6.606
I/
20+ 4
1.36
[2. 48]
4.49
[5. 255 B ]
[5. 51]
5.25 n
7
20+24
1.82,,
[2.42]
4.64 n
5.350
[5. 51.]
[5. 16]
^ /3
40+24
2.34
[3.00]
5.392
6. 179.]
6. 528 B
6.665
v
40+34
2.89 n
[3. 46]
5. 702 n
6. 467]
6.851
6. 979 B
>5
40+44
2.66
[3. 459]
[5. 357]
6. 127,]
6. 530 B
6.653
(0-0 ) 2 cos
ft<WO ,'
I} 2
m- '
2.08
2.08
, y-l. l
5. 546 n
5.546
V
L'*i ;i -
2.54
2.54
, ' /i r' 1
5. 396 B
5.396
7
4
2.5 B
2.5
5.776
5. 776 B
m' 3
m' 3
m' 3 , m'*
m' 3 , m /2
m' 2 , m'
m' 2 , m'
m /2 , m'
m n , m'
m", m'
l cos Arg. +(0-0 )2'M)*))P7) / 9; 2 <C' 2 sin Arg.
where C lt C 3 , C 3 , represent the respective coefficients.
cos Arg.
COMPARISON
OF TABLES.
Table XXX. With the aid of the manuscript the source of all the discrepancies indicated
by brackets has been traced. Coefficients in parentheses are functions of coefficients in paren-
theses in Table XXVII.
Table XXXI. The function was computed by the first of Z 81, eqs. (137), which is more
rigid than the one following it, which v. Zeipel used. Aside from the addition of omitted terms,
the bracketed coefficients are more accurate by reason of the errors in v. Zeipel's Table XVIII.
Table XXXII. The computation was performed according to Z 82, eqs. (139) and (140),
in place of eq. (141) which is less rigid. Besides the discrepancies due to the addition of omitted
terms, four bracketed coefficients are of opposite sign. These discrepancies may be due either
NO. 8.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 135
to a numerical error or to the number of terms included. The remaining discrepancy is due
to slight inaccuracies of v. Zeipel's computation.
Table XXXIII. The discrepancy in this table follows from one in Table XVIII. Third
degree terms in Table XVIII were not integrated because, in the aggregate, they amount to
very little.
Table XXXIV. Our table is more extensive. Second degree terms are, however, not
complete, for they do not include second degree terms in
[%] cos + [z 2 ] sin e
The discrepancies are of no importance.
The integration of eq. (146) is best performed individually for each planet. The analytical
developments are as follows:
The differential equation can be written
By a change of variable
, d * . .=1+0(0)
\ 2 /
Writing
we have Z 96, eq. (152), in which the last term can be neglected.
For a given planet the factors w, TJ, j* and the argument J are known constants. There-
fore 1 +0 (t?) can be expressed as in eq. (153), as a Fourier series of sines and cosines of mul-
tiples of 2#, in which the non trigonometrical term is designated by a.
Expressing eq. (153) in terms of exponentials and solving for d ( -~s [n'te] j by the expansion
of {1+0 (#)}"', and reintroducing the trigonometric functions, we have the equation
following eq. (153), in which the nontrigonometrical part is taken outside the brackets as a
common factor. The brackets in this equation do not have the special significance which
they have had previously.
The variables e and t> are now separate and the integration can be performed. Trans-
ferring the common factor to the left-hand side of the eauation. performing the integration
and adding
n
as the constant of integration, we have the argument expressed as a function of t> in eq. (154),
where is defined by eq. (155).
The reversion of the series gives # as a function of . We have by eq. (154)
where 2Cis & small quantity. Given
z = w + <*0(z), where tx is small,
we have, by a theorem of Lagrange,
By means of this theorem eqs. (156), (157) can be derived, where it is to be noticed that
(C~ 2 C + C/ ) k an approximation for ( ). In our developments we have used ( ).
136 MEMOIRS NATIONAL ACADEMY OF SCIENCES.
If in Z eq. (155) we add and subtract/ -<re [n'dz']]
\2 V
1
f Am i 7? 2\
, O 7* {*** "T~ J-'n I y
r= 2 7 '
C i
Substituting this value of f in eq. (156),
[Vol. XIV.
TIDJ! Ji Oj
iifii JTTOV
+ Series
i :nij 7 LtfHfi 'it't'T -!-:xi.t.!<? ;;<:; !
Substituting the last equation in eq. (145), we obtain Z 98, eqs. (159), (160), and (161). In
2
eq. (160) the factor (s c) is an approximation for - ( ) ; in our work we have used the latter.
2
Since [ndz] t is the series in eq. (156) multiplied by the factor -r
Table XXXV. With the exception of the two coefficients under the heading w*, all the
bracketed quantities are functions of other coefficients in parentheses or brackets, or they are
functions of additional terms. The two coefficients excepted seem to be in disagreement through
some numerical error by v. Zeipel.
Table XXXVI. Since the mass factors have not been kept explicit, it may be well to remark
that only the zero degree term of third order has been included under the heading w 2 .
The bracketed quantities are numerous. Aside from the accumulation of discrepancies
already discussed, the disagreements are to be attributed, in general, to the relative extent of
the computations. It is found from computation that as the number of terms included in a
product is increased the resulting coefficient for a given argument is numerical!}' larger. For
the most part our values are larger than v. Zeipel's. Hence the discrepancies are explained by
assuming that our computation is more extensive. On the other hand, the function is com-
puted much more accurately than is necessary, and many of our disagreements are less important
than they appear to be.
Table XXXVII. The comparison of Tables XXXVII is similar to that for Tables XXXVI
with the exception that our values are not, in general, numerically larger. Some are larger
and some are smaller. Below are brief tables showing to what extent we used the necessary
series. The 0, 1, 2 signify the degrees of the terms included.
1-1
to-'
IP 1
w*
w
w'
w*
w o
w
to
1
1
1
1
1
1
2
2
2
2
2
m'
m"
in!
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
- coe c)W-[(l-ecoee)W]}
137
MLI
-
w-
V
V
V
w>
ii
If!
e
V
w>
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
m'
m"
m"
Table XXXVTIL All the bracketed quantities probably contain only the accumulation
of the discrepancies in Tables XXIX6, XXXVI, and XXXVII. This is a very important
table, and it is from differences in 9 (tf) that the perturbations may be expected to differ most.
PERTURBATIONS OF THE RADIUS VECTOR.
TIT 1 *
If Wand A are tabulated and the computation is performed in duplicate, it is not necessary
3
to make the long developments and the auxiliary tables in Z 6, 99-114. For this reason the
formulae in 6 have not been checked and the list of errata does not cover this section.
The essential formulae are given in Z 99. By Z 7, eq. (36),
In order to parallel the form of ndz, we write
where (flt + 0, + 0,) is given by Z 93, eq. (150).
Hence the computation proceeds as follows: the perturbation is computed by eq. (36),
the argument is replaced by #, and a corrective term which is the product of (0i+0, + s )
and the derivative of the function with respect to # is added. The perturbation v is then
expressed as a function of #. It is tabulated in Table XTJTT.
Table XLHI. If there are no errors of calculation in the construction of the table, all the
discrepancies are due to the accumulation of other discrepancies previously discussed.
The perturbation v =/(0) includes
M
W-*
te
w>
-
m*
Iff!
1
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
m'
m' 2
where the tabulated numbers signify the degrees of the terms included and where only TF, and Ej
are inclusive of third degree.
138
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XLIII.
Logarithmic.
[Vol. XIV.
Unit-1".
Cos
r
ur
HT'
w"
.w
to"
8.72
[9. 88 B ]
1. 6349
2. 1070 B
2. 2333
7 2
9.80
[0.212,]
2.759
3. 4922 n
V
8.9
9.23
2.937
3. 6295 n
?
2. 937 B
3. 6295
t^
J
9.66 B
9.78
3. 1136 B
3.8440
\\*
2t
0. 556 n
1.204
3. 2111 B
3. 7970
1
20+ J
0.504,,
2. 3472
2. 456 B
2. 686 B
3. 4735
rfi
20 + J
0.997
, 1.711 n
3. 6559
4. 3103 B
1*
20 + 4
0.438
1. 220 B
3. 3654
4. 0763 B
j 1 r,'
20 + J
3. 6975 B
4. 3810
1)
20 +2 J
0.438
2. 952 B
3. 2529
[3.0689 B ]
3. 3979 B
V
2J+2J
0. 732 n
1.497
3. 2410 B
4. 0643
11"
2rf+2J
0. 772 B
1.589
3. 4136
4. 0723
X
2i>+2J
3.9048
4. 5649 B ]
4.9303
J > J ' Ji f'bt '- '
* V
2i?+3J
0.505
1.344,
3. 4757 n
2. 783]
&
2i>+ A-S
9.33 n
0.15
2. 938 B
3. 530]
3V
23+2J-2
9.20
0.10 n
2. 0251
3. 2961 B
,;-
+2J
4,?+3J
8.9
9.75 n
1. 2819 B
[1. 5024]
3. 5514
3. 7885 B
3. 6173,,
4. 1394]
3. 8147
4. 3110 B
it'll M
/
4^+44
4V+3J-2
9.98
[1. 1342 B ]
9.64 n
3.4007
2.305
3. 9091 B ]
2. 542 B
[4. 1480]
[2. 749 n ]
,'3
6^+34
0.436
1. 220 n
4. 2675
4. 7993 B
ft*
6^+4J
1. 125 B
1.862
4. 6479 B
5. 2324]
?V
6tf+5J
1.198
1.947 B
4. 5397
5. 1768,]
& :IT
? 3
6J+6J
0. 732 B
1.508
3. 9457 B
4. 6328]
P *'
60+4 J-2 1
9.20
0.10 B
3. 4099
4. 1710 B
V
6^+5J--T
9.70 B
0.56
3. 2601 B
[4. 0542]
>M'
ie+ <>
3,4878 B
4. 1106
i
^+ t>+ J
8.3 B
2. 2106
2. 7179 n
2.919
; 2
is+ t5+ J
3. 5709 B
4. 2261
% 2
ie+ ?+ 4
3.4507
4. 1296 n
V
i+ l>+ A
3.5100
4. 1837 B
tt*
}.+ iJ+24
2. 579 n
3. 9270
?'
is+3^+2J
0.08
3. 6873
4. 1471 B
4. 7839
^
4+3t+3J
9.5
3. 5727 n
4. 1511
4. 7545 n
,"
is+5<J+3J
4. 5568
5. 1414,
tr
i+5<5+44
4. 7261n
[5. 4067]
/
i+5<>+5J
> a JOTIO ti<
4. 2862
[5. 0418n]
/
if+5<>+4J-2'
UoiUlflUil
3. 2570
4.0005 n
y"
-*+ <>
1. 086 B
2.7090
3. 3467 B
3. 7098
i|
-<:f+ I>+ J
0.88
2. 1967 n
3. 0952
3. 5836 B
V 2
-- U+3<?+ 4
2.514
4. 1049 n
7V
"
-,:J + 3tf + 2J
4. 0853
[3. 9122]
-i +3i5+3J
3. 8341 B
[3. 8U8]
f
-it+3iJ+2J-J
2.416
3. 6926 n
ij
e
9.62
0.58 B
2. 143 B
2. 682 2. 9151 n
V
+ ^
9.04 B
9.9
2.061
2. 666 n 2. 9477
tV
+ 2l>+ J
0.444
1. 1661 B
3. 0588
3. 8035 B
[4. 2554]
e+2i?+2J
9.487
2.]744 B
2. 7280
2. 972 n
2.976
>! 2
+2tf+2J
0.344n
1.1143
2. 692 B
[3. 5334]
[4. 0772 B ]
I"
+2tJ+2J
0. 025 n
0.828
2.634
3. 0726
4. 0416 B
f
+2^+2J
3. 1265
3. 8806 B
4. 3473
W'
+2tf+3J
9.98
a sii.
2. 873 B
3. 1697
[3. 5856]
tr*
+4i?+2J
1.105
L89W
2.864
4. 3477 n
r
+4^+3J
8. 8 n 0. 398
2. 8000 n
3. 5327
4. 0065 B
4. 3207
,Y
+4^+3J
1. 260 n 2. 083
3. 0931
4. 4160
,"
e+4<+3J
3. 8375
4. 0446 B
J 1 >>'
+4^+3J
0. 267 1. 15 B
3.9421
4. 6972 B
j)
e+4tf+4J
9.19
0. 248 B
[2. 6356]
3. 4317 B
3. 9469
4. 2558,
^ 3
+4^+4J
0. 774 1. 66,,
3. 0934 B
[3. 7866 n ]
,,"
s+4i>+44
4. 1154 B
4. 5547
f.
+4tJ+4J
0. 455 B 1. 32
3. 8518 n
4. 6436
Jv
+4^+5J
3. 7579
4. 3244 n
No. 3.]
Logarithmic.
MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
TABLE XLIII Continued.
139
Unlt-1".
Cos
-
r
*
.
v>
fi
e+4t+3J-J
3.0030
3. 8869,
I 3 n'
, 4i , , 4j_^
2.4425
1 85,
'/
r
g-j-6(?~l~4^
9.98,
0.480
3. 5016,
4l 3723
4. 9952,
-{-6i?-f~5J
+6t?+6J
e~}*6i?-(-5^ ^
0.296
9.95,
8.5,
0.823,
0.538
[9. 15]
3. 6369
3. 1685,
2.114,
[4- 5582,]
[4. 1334]
3.0881
5. 2093
[4. 8131,]
[3. 7886,]
3 i"
t-f"8^-|~5^
4. 2554,
4. 9349
!*
cr
f-j-8i?-|-7^
1.320
1.228,
2. 152,
2.093
4.5657
4. 3995,
5. 3010,
5. 1827
,'
P^-gjj-Lgj
0.648
1.54,
3.7543
4. 5812,
/
^j-gjj -i- j2
3. 2818,
4.1442
A
s-\~o^-\-7^ ~~ 2
3. 0763
3. 9759,
- t+2d
0.305
PL 1007,]
2.912
3.4958,
3. 8151]
j/
- ;JgJ^
0.490,
0.117
[1. 3330]
2:288,
[3. 7273]
3. 2375,]
4. 3119
3. 7892
a
~f"2l?-j- J
9.04
9.96,
2.636
3. 2817,
3. 6568
B
~f"4i?-{- J
3. 2197
3.9650,
- e+4t>+2J
1. 146,
1.89
3.0204
4.2441
n 2 j/
f~|-4i?-l-3^
1.005
1.78,
3. 5247,
4.0012,
B
f-|-4(?-t-4i
0.290,
1.15
3. 1793
2.982
1 K
-{~4tJ-}-2J ^*
3. 2486
4.0585,
- f+4tf+3J-J
9.98,
0.8
2.957,
3.8580
n
| + ,j+ J
9.0
2.3363
[3.0704,]
[3. 5111]
,;
| + 0+2J
9.5
1.500
2.3585
3. 1842,
|+3*+2J
2.779
3. 7820,
If+SiJ+SJ
9.28
2. 1614,
3. 0257
3. 6491,
i) 1
j_j_3 ( j_i_3j
1.32
2.966
V
*+3iJ+3J
3.3450
4. 1111,
j *
|+3i>+3J
3. 2309
4.1965,
|+3<?+4J
3. 2994,
4.1520
'
f+5<>+4J
1.017
3. 1617,
4.1967
5.0160,
-
5+5<>-i-5J
0.88,
2.9688
4. 0380,
4. 8781
fl"
4.0855 n
5. 2422
5"!'
^+7<)+6J
4. 1991
5. 3823,
f+7iJ+7J
3. 7114,
4.9188
f
| ^_7,j-(-6J 2"
2.615,
3.8317
y ,'
3. 2411
3. 7872,
.2
' + tj+ J
2. 819,
3.4476
y 3
-*H- tf -I 1
2. 9181,
3. 4813
B
2
2.364,
3. 0737
TJ Ij
2+ J
2.624
3. 3489,
U/z
2 S + 24
2.207,
2.978
' X *lt
y 2
2+ J+^
2.620,
3.2765
,j
2+2<?+2J
9 8,
1.63
2. 362,
2.873
r/
2e+2i>+3J
9.5
1.796
2.303,
2.1007
2 +4<J+3J
1.92,
2.700
2j+4!y+4J
8. 7 [8. 8]
1.5802,
2.4158
2. 9867,
gl
2+4tj+4J
2.330
3.1764,
jj'2
2j+4^+4J
3. 1079
3.9008,
/*
2j+4tj+4J
Hi t):'f?>l;i|'ni
2.736
3.6809,
. . p V .,}
" .
5 T
2j+4^+5J
2. 9881,
3.8425
-
il+S+ej
9.64
[0. 53]
[0. 36*]
2.652,
2.4419
3.6204
3. 4512,
4. 3279,
4. 1892
l"
2+8;>+6J
3. 6135,
4.6784
n *'
2+8i>+7J
3. 7124
4. 8075,
'
M
2f+8^+8J
.$ i'i >i
au jV* '>/J'l
3. 2109,
4. 3338
i '. in IllSilOf
y 2
2s+8d+7J-2
2.068,
3.2092
yxjiii jd r>
| c - +5t j + 5j
9.3,
1.140,
2.0056
2. 5727,
,j'
+7i)+64
0.5,
2. 2749,
3. 2377
3.9184,
i
|+7tf+7J
0.3
2.0542
3.0565,
3. 7710
j+7<>+7J
8.1
0.43,
1.346
1.959,
140
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE XLIII Continued.
Logarithmic.
[Vol. XIV.
Unit-1"
Cos
ur
ur
KT'
UJO
to
.
(t>-t )sin
99'
^
9.66
0. 810 B
2. 7559
3.3840 n
3. 6946
9 /
2t>4- 4
9.79 B
0.54
1
20+2J
9.92
[0. 63]
1
I
9.801 B
0.425
2. 5970 n
3. 1493
3. 4158 B
I 3
1. 075 n
2.045
3. 5201 B
4. 3751
II' 1
e
1.640 n
2.514
4. 1066 n
4. 8961
ft
1
1.063
1. 916 n
4. 1066
4. 8961 B
5. 4076
1'
+ 4
9.36
0. 471 B
2. 4824
3. 0830 B
3. 3936
1\
+ J
1.565
2. 456 n
3. 9671
4. 7890 n
t+ J
1.543
2. 341 B
3. 8942
4. 6760 B
? ';,
+ J
0.87 B
1.75
4. 0705 B
4. 8759
5. 3965 B
t+ 2J
1. 441 B
2.273
3. 7192 n
4.5456
f i f
t+ I 1
0.42
1.36 B
3. 7799
4. 5819 B
5. 0998
e+ ^+2
0. 695 B
1.585
4.0417 B
4. 7827
5. 2543 B
,a
j+4,>+4j
9.59 B
0.45
i'
+4tf+3J
9.46
0.34,
1
2t+2*+2J
9.45
[ H]
*
2 -j-2<?-)-3J
9.32 B
[O^CM]
9V
- H- J
1.255 B
2.149
3. 6240 B
4. 4615
(l> 1> ) 2 COB
,
C
9.25
0. 117 B
''
f+ J
9.12 B
0.02
TO' 2
TO' 2
m' 2 , TO'
TO'
m'
TO'
cos Arg.+(tf-i? )^u.-r / P7j / 9; 2 C 2 sin Ar
where CD C 2 , C a represent the respective coefficients.
PERTURBATIONS OF THE THIRD COORDINATE.
Arg.
For the third coordinate the developments are limited to perturbations of the first order
and of the first degree with the exception of some secular terms of second degree. We can
therefore use osculating elements in this section, and use 6 and # without distinction.
By Z 8 eq. (39), 41, eq. (83) and 8, eq. (41) the equations Z 115, (192) are given, in which
2 is defined.
Since
dS_SS SS ^ = 2
de de + 3d de
By Z 9, eq. (45) we have, with sufficient accuracy, Z 115, eqs. (193). Within these limits,
dO w
Substituting this relation in the above equation and in eq. (192) in turn, the differential equation
to be integrated is (194).
Since F, 6, H are power series in w, it is evident from eqs. (192) that
j O
^=
where
NO. s.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 141
Therefore, eq. (194) becomes
Comparing the coefficients of like powers of w on either side of the equation, it is evident
that the integral must be of the form
Substituting this relation in the preceding equation and equating like powers of w, the system
of equations (195_,) (195 t ) follows.
Within the extent of the following developments one more equation should be written by
analogy.
dW
This system of equations is integrated in a manner similar to that for -5- (see p. 81). Each
equation is broken up into two equations, one a function of e and one independent of e. The
differential equation (194) is then replaced by eight differential equations, the integrals of which
can be obtained in the order,
S_,, GS.-DSJ), [SJ, (S^-DSfJ), [SJ,
As in the case of -j , the condition is imposed that
The equivalent equations are (196)-(200).
dW"
A comparison of the differential equations for (S< [S<]) with the expressions for , *
dW "
s- 5 leads to an analogous form of integration for certain terms. Within the extent of our
developments,
and
1
-(l-cos) . -c -!- cos
dW " d W "
take the place of ^ and ^77 respectively. Without change of notation for the third
coordinate, (S-[S]) is given by eqs. (201), (202), where P, G, Q are computed from F, G, H in
Tables XII-XTV, by means of eqs. (118) and (119). The coefficients P, G, S are tabulated in
Tables L to LIT.
The function [S] is obtained from the integration of eq. (203). A constant of integration
is added, which is the same in form as Hansen's constant of integration for the perturbation of
the third coordinate, namely,
c,(cos <f> e) +Cj sin <f> Z eq. (204)
where c l and c t are undetermined.
By eqs. (192), the pertubation '. is derived from
COS c
.
i cos i
The perturbation comprises the computed value of eq. (202), the trigonometric sine series given
by Tables L to LII (which can be written by inspection with the aid of Table XV6), the series
forming Table LHI, and the constant of integration (204), in all of which
142
MEMOIRS NATIONAL ACADEMY OF SCIENCES. tvoi.xiv.
TABLE L. Unit 1".
n
l
2
3
4
5
J' 1 . (n+l.-n+l)-Hr /
+ 52.7
+ 96.0
+ 57.0
+ 33.8
+ 20.1
+ 12.0
*io(** 1. 1+1)+^'
+158. 2
- 285.0
-101.4
- 47.0
- 24.0
?Vo( n +l- n 1) '
-158. 2
-191. 9
- 95.0
-50.7
- 28.2
- 16.0
*Vo(-l.--l)-*'
- 52.7
-191. 9
- 285.0
+ 140. 9
+ 48.1
b
J*,. (n+l.-n+l)+^ /
-201
-352
- 253
-176
- 119
- 80
M
J" 1 . (n-l.-n+l)+^
-812
+1495
+594
+ 305
+172
5
/,. (n+l.n I);:'
+812
+897
+ 498
+297
+ 183
+114
(
JWn-l.-n-l)-,:'
+201
+513
+ 355
-1478
-439
TABLE LI.
Unlt-l".
<3 . (n.-n+l)+K'
26.37
- 47. 98
- 28.50
16.91
10.06
- 6.02
Go^n.-n-l)-^'
+ 79. 10
+ 95. 96
+ 47.50
+ 25. 36
+ 14. 09
+ 8.02
5 1 . (n+l.-n+l)+T /
+ 90.3
+ 112. 3
+ 58.5
+ 29.0
+ 13.6
+ 5.8
<5,. (7l 1. 71+1)+^'
+ 530. 8
+ 720. 8
+ 468.9
+ 311. 7
+ 207. 7
+ 138.0
G,. (n+l. n 1) IT'
- 124.2
- 120. 5
- 53.3
- 21.8
- 7.4
- 1.2
G,. (n 1. n 1) T/
+ 609. 9
- 1549.1
- 674.1
- 369. 6
- 219.0
G . 1 (n.-n+2)+r /
- 162.4
- 211. 6
- 103. 8
- 47.7
- 19.7
- 6.4
<2o-i( n - n)4V
- 166.5
- 352. 6
- 298. 2
- 229.0
- 167.2
- 118.3
<?.,(. n) it'
+ 166.5
+ 96.7
+ 13.2
- 14.4
- 20.7
- 19.3
G .i(n.-n-2)-7: /
+ 1825.5
+ 881. 4
+ 516. 6
+ 321. 2
+ 204. 1
G . (n.-n+l)+ff'
+ 100.4
+ 176.3
+ 126. 8
+ 87.8
+ 59.6
+ 39.9
Go-ofa- 1) *'
- 406. 6
- 448. 6
- 249. 2
- 148.6
- 91.5
- 57.2
G,. (n+l.-n+l)+w /
- 432
- 592
- 370
- 218
- 122
- 64
^ i -o(. n ~ 1 ~ n H~ 1 ) +f'
-2047
- 3183
- 2412
-1811
-1342
- 982
Gi.o(n+l. -n-\)x'
+ 718
+ 821
+ 440
+ 225
+ 107
+ 44
o
Gi. (nl.nl)7^
-2401
+12134
+4939
+2788
+1744
1
G . l (n.-n+2)+x'
+ 693
+ 951
+ 568
+ 314
+ 158
+ 68
Gjo.^n. n)+7r'
+ 893
+ 1773
+ 1607
+1356
+1089
+ 844
G .j(7i. n) w 7
- 893
- 747
- 254
- 27
+ 68
+ 98
gj / _ n _9'i_ /
-13263
- 5889
-3549
-2336
-1586
1 ' ^
TABLE LII.
Uuit-l".
F . (n.-n+l)+w /
- 79. 10
+ 142.49
+ 50.72
+ 23. 48
+ 12. 03
fl . (n.-n-l)-*'
+ 26. 37
+ 95. 96
+ 142.49
- 70. 45
- 24. 07
#,.0(71+1. -n+l)+jr'
- 609.9
- 528.9
- 231.4
- 108.8
- 52.7
- 25.7
Hi. (n-l.n+l)+x'
+ 124. 2
+ 528. 9
+1121. 6
- 897.4
- 365. 9
ffi. (w+l. wl)t /
- 530.8
+ 551.7
+ 166. 9
+ 64.3
+ 26.5
tf^n-l.-n-l)-*'
, ;i yrj 90.3
- 312.4
- 421.4
- 572. 6
- 967.8
H . l (n.-n+2)+7:'
+1057. 8
+ 311. 5
+ 111.3
+ 39.4
+ 11.6
S tl .i(n.n)-\-x /
- 166.5
-1057. 8
+1145. 2
+ 501.5
+ 276.
H O ,,(.-)-K'
+ 166.5
+ 290.1
+ 71.9
+ 62.1
+ 44.9
H . 1 (n.-n-2)-7r /
+ 162.4
+ 608.5
+ 881.4
+1551.
- 1020.6
H . (n.-n+l)+x'
+ 406. 6
- 747.8
- 297.2
- 152.4
- 85.8
H . (n.-n 1)-*'
- 100.4
- 256.6
- 177.8
+ 739. 1
+ 219. 8
^,. (n+l.-n+l)+^ /
+2402
+2483
+1362
+ 740
+ 406
+ 222
I
Zr,. (n-l.-n+l)+,r'
- 717
-2483
-4550
+8048
+ 3120
y
F,. (n+l.-n-l)-^
+2046
+ At c y
_I_1 91 A
-4336
1 T API
-1408
_i_l CMYI
- 697
11 -\ Of*
298
1
fi Q \(n _ n +2)+^ *
<6&
-f-U14
-3908
-rJ.4ol
-1705
-piyui
- 753
-plloO
- 325
- 126
8 9 . l (n.-n)+* f
+ 893
+3908
-9529
-3936
- 2233
n^.^n.n)!^
- 893
-1855
39
- 287
- 270
^ . 1 (n.-n-2)-^
- 693
-1987
-2363
- 310
+13643
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE LIU.
} Unit-l"
143
and by eq. (193),
Sin
UP 1
w"
10
v"*-
^+40+3J-n'
- 25. 36
+ 123.2
- 281.8
^
40+3J-IF
-J+20+ J-U'
<j>+26+M+U'
<!>+26+ J-U'
f+W+M-U'
+ 50.7
- 816. 8
- 521.8
+ 432.9
+ 129.9
- 246. 5
+ 3636
+ 2851
- 2034
- 861
+ 563.6
-8548
-7663
+5237
+2770
if
4-26 +n'
<!>+28+24+n'
<!>+28+2J-n'
0+60+4 J-n'
- 649.4
+ 596.4
- 26.5
- 214
+ 3096
- 2916
+ 494
+ 1236
-7475
+7216
-2266
-3395
(0-0 ) cos
4+ J+U'
+ 191. 93
- 705. 2
+1302. 6
y
4+n'
- 383.8
+ 1410
-2605
1)'
4+ A+H'
4- 4-n'
+6584
-5312
-40060
+29610
ril'
J>+ 2J+U'
<!>+ n'
4- n'
-5742
-6024
+6024
+36970
+38180
-38180
1"
<<,+ A+W
$+ J-U'
+6584
-1656
-40060
[+11860]
f
4+ j+n'
-3002
+18970
m'
2
By inspection it is clear that the periodic part of S is of the form
2 Up. q i)Pr]'* sin A
and the secular terms are of the form
o'A'^ cos {(A-t
w
. T) 17,. COS A
Expanding cos {(A s) +s}, and collecting coefficients of sin and cos e, the secular terms can
be written
nt{ K^cos e e) + Kj sin e}
where
IT, = I U p . q ijPi)'i^ cos (A E) - 4 C/j.o cos A
Introducing this notation, the perturbation can be written in the form of eq. (205).
The coefficients U p . q are given in Table LIV. K, and K 2 , which are constants, are tabu-
lated in Tables LV^ and LV n , respectively. For a given planet the factors and arguments are
known. Therefore T^ and K^ reduce each to a single numerical quantity.
Since the Bohlin-v.Zeipel method is based on the fundamental principles of Hansen, the
constants of integration are determined by the condition which must be satisfied when the
144
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
perturbations are developed on the basis of osculating elements, namely, that the perturbations
and their first derivatives shall be zero at the time t = 0. The relations to be satisfied are
u =
-0
dt~".
and the following equations are equivalent relations :
Logarithmic.
Unit-l"
Sin
M
w'
W
1) f
1 x
- 4-n'
-n'
29+ 4-n'
49+34 -H'
49+24 -n'
1.705
3. 0621 B
2. 8235
2.2831
3. 1591,,
3. 2462
3. 7258
3. 5528
2. 8483 B
3. 8608
3. 9166 B
V
j + 9 -n'
i+ 9+ 4-n'
i+39+24-H'
J:+59+44 n'
'II
1 1 !
3. 2112 B
2. 5875
2. 2787
3.3155
3. 0779 B
3.8544
3. 4153 B
2. 6304 n
3. 5865 B
3. 3972
'<;
\e 9 24 n'
-}e+ 9 -n^
_ j -j-39+24 n'
3. 1158 B
3. 1493
2. 3242
3. 3863
3. 3532 n
3. 7378
3. 7544 B
3. 0060 n
4. 1833 B
4. 1452
i
-)-29+ 4 n'
+29+24 -n'
+69+44 -n'
+69+54 -n'
2.6364
1.423n
1. 4042 n
2. 3306,,
2. 1137
3. 3704 B
2.706
2. 1720
3. 1922
3. 0138 n
3.8423
3. 4014 n
2. 6339 B
3. 7582 n
3. 6101
T\
- -29-34 -n'
- -29-24 -H'
- - 4-n'
- +29 -n'
- +29+ 4-n'
2. 7175
2. 7756 B
2. 8125
2. 9121n
3. 4858 B
3.5070
1. 6810
3. 4427 B
3. 4958
3.9484
3. 9456 n
2. 2463 B
3.7846
3. 8338 n
'* '
$+39+24 -n'
$+39+34 -H'
$e+59+44 -n'
f +79+54 -n'
2.6058
1.760
1. 7510 n
2. 9120 B
3. 5312 n
1.82 B
2. 8113
4. 0813
'';
-f*- 9-24-H'
-$+ 9- 4-H'
-$+ e -n'
i ''A '
2. 8673
2. 9620 B
2. 0569 B
2. 9275 B
2. 9702
3.8458^,
3. 9124
2. 7932
3. 4708
3. 5487 B
v
2+49+34-n'
2+49+44-n'
2+69+54-H'
1.640
1.617
1. 206 B
2. 731 n
2. 340 n
2. 2110
TJ
-2e-49-54-n'
-2i-49-44-n'
-2-29-34-H'
-2 -24 -n'
-2j - 4-n'
2. 4012
2. 5241 B
1. 5290 n
2. 3174 n
2. 3514
3. 3634 B
3.4544
2. 3210
3. 0558
3. 0737 B
m'
u
.=2Ur,. a i)P-n'9a\n A+nt{K,(coa e )+JsT, sin }+c,(cos t e)+c, ain E
I COS I V V 1
7 IT
No. 3.]
>>!
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABLE LV a .
145
Logarithmic
Unit- 1".
OH
w
V
*
1
j-n' '
j+n'
^+n'
4+n'
j+n'
2J+n'
2. 9180 B
1.9821
2.8035
3. 5175
3. 1764 B
3.4580 n
3. 7732
2.5473,
3.7182,,
4. 3017,
3. 9772
4.2668
2.8138
m'
/*/* cos Arg.
. -A. i TABLK LV '- ;o ;
Logarithmic
Unit-l".
i j-.rl* if e.*
Sin
tr
u
w
.
' i .> 1 'yls III *WJ
, .,.,() ;f.tj|,,
f.. viioiti -fiKt
V*
j n'
2.9180
3. 7732,
^
n'
3.7799
4. 5819,
4+n'
1. 9821,
2.5473
2.8138,
>i>t .(i<j(jfnr!''!>
'V
j+n'
3.7744,
3. 5175,
4.5420
4. 3017
1 v'
2J+n'
3.4580
4.2668,
i r '. *- - . i , r f
j"
j+n'
3.1764
3. 9772,
r,,fT .]ioll;)fl.l
n'
F! 1 ". ' :'.' ' >' \
ITj = SvPTpy'i]* sin Arg.
1^
i;Pj;'9 sin ^4 +7J.K, (cos s e) + K, sin e +c,(cos e e) -fc, sin e
r cos i ~ M
By eq. (205), at the date of osculation,
/ = ft n = fj-
-
w
, , ' i
c,(cos e e) +c, sin s (A.)
t COS t
By Hansen, 1
d( u \ d( U \dS
d\t cos t/ = d^V cos i) d<f> U
in which v. Zeipel's notation is adopted.
dS
xi_ . .,
tne derivative, -, contains the constants
<t ^ongiateafT
>iiT *.i JT
From the various parts of S, enumerated above, *jj can be computed. Since S contains
the constants of integration
c,(cos <f> e) +0, sin ^
.'ii -tMl . L' 'i9<i yjininqolsv^b aniwollol 'idl i<>}
IF! v wMi/ju-. -noil tJOTim..-> od IIAO -T;i-wn-(> -dl
c, sin e + c 3 cos e
The solution of eqs. (A) and (B) gives c^ and Cj. But there is a better way of deter-
mining the derivative of the perturbation. The exposition of this second method is
postponed until a particular example is considered, for the perturbations are not yet in a form
which leads to the development of the equations.
1 Auseinandersetzung einer iweckmassigen Methode rur Berechnung der Absoluten Stomngen der Idcinen Planeten, Erste Abhandlun, 5 5, p. S
110379 22 10
>,-ib 1.,'; ITO! :! ll/-.''''
146 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xnr.
COMPARISON OF TABLES.
Tables L, LI, LII check satisfactorily.
Table LIII. With one exception, the agreement is satisfactory. The bracketed coefficient
contains a misprint in sign in v. Zeipel's table. That it is a misprint is evident from Table LV t ,
in which the correct sign is given to the corresponding coefficient.
The terms included in the last column are computed from the additional tables,
2 , XlVifl 2 and from first degree terms in Z 116, eq. (200). The latter part, namely,
e cos .
is added to both eq. (200) and eq. (203).
Table LIV. Our table is more extensive. The one bracketed quantity includes an addi-
tional term from Table LIII.
Tables LV I( LV U check satisfactorily.
CONSTANTS OF INTEGRATION IN ndz AND v.
The constants in . were treated in the preceding section by the familiar Hansen method.
cos i
It is the purpose of this section to modify the similar treatment of the constants in the per-
turbations ndz and v so as to incorporate them in the elements a w e , ic w ^ - Through the con-
stants of integration, the constant elements, which have been used from the beginning without
definition, are to be explained.
Since the group method of developing perturbations is built upon the fundamental prin-
ciples of Hansen, his conditions for the determination of the constants of integration must be
fulfilled. These conditions depend upon the choice of initial osculating or mean elements.
Osculating elements are used here. The corresponding conditions are that the perturbations
and their first derivatives, at the date of osculation, (< = 0), shall be zero.
Consider the relation of the constants of integration to the elements. There are two con-
stants in each perturbation since the differential equations are of the second order. The con-
stant added in the first integration is a velocity; the one added in the second integration is a
displacement, or, a perturbation. Now, recalling that the position and velocity of a body for
any time t can be transformed into the constants which are ordinarily called the elements of
the orbit, it is evident, by analogy, that a displacement of the body and the velocity of the
displacement can be transformed similarly into changes in the elements. The four constants
in n$z and v are related to the four elements, a, e, TT, c, defining the shape and size of the orbit
and the position in the orbit, and the two constants in the perturbation which is measured perpen-
dicular to the plane of the orbit are related to the elements fi, i, which determine the position
of the plane of the orbit. It is possible therefore to modify all six elements, but it is v. Zeipel's
preference to make the transformations only for the first four constants.
It is not necessary to compute
ndz v
k> eJificj euorutv 'fi mo
dn8z dv t =
Jflfjj- <:<)> )ti)
de de
for the following developments perform the transformation analytically, and the changes in
the elements can be computed from auxiliary functions.
Let a , e , x , c be osculating elements ; let a, e, K, c be the osculating elements modified
by the constants of integration in the manner indicated above.
For undisturbed motion,
-e sin = c + 7i < ll+^>
<sKv-7M,) = -\/nrz *9 i
r cos (v 7r ) =a (cos e e ) r sin (v JT O ) =a Vl e 2 sin E
Hansen's choice of ideal coordinates demands that the coordinates and their velocities
shall have the same form of expression for disturbed and undisturbed motion. The ideal polar
NO. 3.] MINOR PLANETS LEUSCHNER, GLANCY, LEVY. 147
coordinates are designated by E or /and f. The relations which are analogous to the above are
!l!JMV4 i!) HO
tg(v -JT O )
F cos/=a (cos e-e ) f sm/=a o yi -e e 2 sin I
f=vn^ r=f(l+v)
These are the equations for motion in the orbit based on constant osculating elements and
appropriately determined constants of integration.
If, in place of osculating elements and Hansen's ndz and v, v. Zeipel's elements and the
corresponding perturbations are used, the equations are the same in form. In v. Zeipel's
notation e and / take the place of e and /. The omission of the dash over these variables is
permissible, since the physically real values, with which they might be confused, do not occur
in the theory except for the date of osculation, where the subscript zero is added. It is to
be noted that, through v. Zeipel's choice of elements, the coordinates and the perturbations
have values which are numerically different from the Hansen quantities of the same designation.
Let the time be the date of osculation and denote the true coordinates by , v , r,. Then
the preceding equations for undisturbed motion become Z 121, equations (206), (207), and
Z 125, equation (230).
Let the disturbed eccentric anomaly and radius vector (e, r) be e, and r v respectively.
The relations for disturbed motion become Z 121, equation (209), and Z 122, equations (210).
The first derivatives of these expressions are given by equations (208) and (211), respec-
tively, and the time rate of is given by the equation following (209).
The solution of the four equations (210), (211), with the aid of all the others, determines
the four unknown constant elements, a, e, n, c, or, better, a a , e e , r K O , and c.
The fact that the adoption of the new elements in connection with the perturbations ndz
and v, as developed in the preceding sections, is equivalent to the use of osculating elements,
follows from the simultaneous solution of the equations for the disturbed coordinates and their
velocities and the corresponding equations for undisturbed motion.
The method of calculating c from the equation
c = , - e sin , - ndz
is given in the example, page 18.
After many laborious transformations the other three unknowns are expressed in terms of
familiar functions in equations (233)-(236). In the verification of these equations slight differ-
ences in the numerical coefficients of certain unimportant terms were found. The magnitudes
of these coefficients depend upon the number of the terms included in making the transforma-
tions. Since it makes little difference whether or not they are included and since v. Zeipel's
values present a more symmetrical form of a later auxiliary function, we adopted his coeffi-
cients.
In the functions x, y, z the arguments and factors are functions of ij, ic, ff v J, 2, where
0, = 2 ki-* sin e,)-^'
but at the beginning of the computation only T; O , r , , , J , J c , the corresponding functions of
osculating elements are known. 1
i There is a confusion of notation in v. Zeipel's developments. In Z 127, equation (238), Ha is denned to be the value of at the date of oscu-
lation when osculating elements are used for the planet, and 0\ signifies tin- argument if the elements a, t, *, etc are employed or by Z 9
equation (43), l
hykt-e, sin )-/
j -l-ii.lV <)\(>l\'
and their di'Terence is computed by Z 127, equation (238).
In the collection of formulae by Z 133,
0- 4 c. c'
This is an approximation for the above equation.
Oi-ic-c'
-j- (n e sin )-# ef
If the secular terms are counted from the date of osculation, the factor (9 ) ought to be replaced by (0 ft).
148 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.
By equations Z (43), (235), (236) and the equations preceding (233), the factor ij and the
arguments J, e w t are given in equations (238) in terms of osculating values and functions of
perturbations, inclusive of first order.
To these should be added
1 i
2 = 2 -j-s+
and 4ljo
-i, cos s sn e n -z cos
where ,1- j
The equations (233), (235), (236), and (238) permit the construction of two tables which
determine w, n or a, and e and TT. From here on the developments differ in form from v. Zeipel's
although they are the same in principle. If v. Zeipel's equations (237) and (239) are used, the
term (x," ijy/') should read
(* 2 " +x s " +x t ")-i)(y 3 " +y s " +y t ")
in agreement with Z 91, line 14.
Suppose that w w has been computed by equation (233) and the argument F has been
introduced. The arguments and factors are unknown.
.1' ')()*:) sjmwnliot 1; ' ''.'* nv/ij} 8* i io -jKM 3;m' f>rlf bn ,-.M-jnt
By Taylor's theorem ~ W =/()J ' FV &u *' ^
df 8f df 8f df
w-*>*=f(>), r, e , J , S )+-^j, +-^jr + ^j0 +^jj +^jj + r .,,^ wj
Inclusive of second order in m', the differentiation is for first order terms.
Substituting the values of Jij, AF, J0 , JJ , JJ from equations (238) and the additional
equations above,
_,, ra A iMa./*/. 5 / 5 /N ! /'*/' V VV
.-/^ / , , *., ^+\f^3 5Jo ^ ;4^ 2+ VH, wiT'.WwSS!
ai i ^3/ 2 / 1 -
,,
"'Jo COS + 1-^9 COS ,l/ Sin -3 COS
The order of calculation is: computation of equation (233), in which the arguments and
the factors are given the subscript zero, differentiation of first order terms, computation of the
second order terms in the above equation, and the additon of these second order terms to the
first calculation.
With some foresight the computation can be simplified. The arguments should be arranged
in groups like the following:
TU f u i /
ihen, for whole groups of arguments,
df _8j_ _d _
~~-
Also for some particular argument in a group, the condition
may be satisfied.
Ko.8.]
MINOR PLANETS LEUSCHNER, CLANCY, LEVY.
149
Finally, by inspection of the arguments, considerable computation can be avoided if
._ A/L a/
The function w w> is tabulated in Table LVL Since it is unavoidably a function of w
itself, the determination of to for a given case must be made by successive trials, the first
approximation being n>=w
Logarithmic.
TABUS LVI.
w tc
Unit -1 radian.
Cos
~
^
-
K*
w
..
4.360
[5. 1966,]
[5. 7767]
r
4.766
6.6599
7. 3732,
7.7492
zr
4.446
7.1194
7. 7572,
8.0553
3r
4.412
6.8442
7.5458,
7.9060
4.484
6.5883
7.3450,
7.7602
5/ 1
6.3437
7. 1490,
7. 6136
7r
5.875
6.7632,
7.3134
I'D
-5r+20 +2J
-4r+20 +2J
4.161,
6.5090
6.169
6.6325,
7.0658
7. 4746,
7.8698,
-3r+20 +2J
-2r+20 +2J
3.19
3.52
6. 8821,
7.0986,
7.6078
7. 6970
7.9975,
7.9394,
- r+20 +2j
5.1420
6.359
7.0722,
7.4480
20 +2J
4.379
7.6355,
8.2144
8.4125,
r+2S +2J
4.856,
8.0894,
8.9548
9.5668,
2r+20 +2J
4.92,
7. 8150,
8.6561
9.2006,
3r+20 +2J
5. 5174,
7.6056,
8.4650
9. 0111,
4F+20 +2J
5.4248,
7.4128,
[8. 2958]
[8. 8561,]
5r+20 +2J
7.2254,
8.1426
8.7346,
7r+20 +2J
[6. 8746,]
[7.8484]
8. 4936,
jf
-5r+20 + J
6.8776,
7.5604
7.8425,
-4r+20 + J
4.582
6.8815,
7.4536
7.5238,
-3r+20 + J
4.674
6.6271,
6. 7816
7. 3174
-2r+20 + J
4.99
6.7985
7.4732,
7. 7966
- r+20 + J
5.4623,
20 + J
4.605,
[7. 1987]
7.8314,
8.1061
r+20 + J
5.0056
8.2964
9.1086,
9.6833
2r+20 + J
4.38
8.0434
8.8316,
9.3296
sr+20 + J
5.6251
7.8458
8.6564,
9.1558
5.5812
7.6603
8.5030,
9.0248
5F+20 + J
7. 4778
8.3544,
8.9050
7r+20 + j.
7.1130
8.0545,
a6668
"to*
4.664
4.n
5.83
r
7.8102
a 6250,
2r
7.7520,
ai242
sr
7.6172,
6.6043,
4r
7.7135,
a2308
fc 1
4r+4ff tt +4J,
7.1862
7.9072,
_3/^_j-40 -j-4j
7.1804
7. 8679,
2f+40 +4J
6.817
7.456,
F+40 +4Jo
8.4680,
8.8822
40 +4J
4.666
[5. 807,]
[8. 0913]
a 8270,
9.2073
/"+40 +4J
a 7850
9.8236,
2r"+40 +4J
[a 5144]
9. 4910,
3f+40 +4J
a 3274
9.3006,
4/"'_j-40 -^4 <
8. 1627
9. 1494,
5r+40 +4J
8.0050
9.0105,
wf
_4f-(-40 4-3j o
7.354,
a 1083
-3^+40o+3jo
7.5708,
8.2084
-~r+40+3Jo
8.8838
9.0548,
40 +3 J
4. 516,
[6.2084]
8.5565,
9.2180
9. 5174,
r t +40 +3J
9.2783,
0.2833
2f +40 +3J
9.0241,
9.9635
3f +40 +3J
a 8480,
9.7850
4r+40 +3J
a 6916,
9.6434
$r +40 +3 J
8.5401,
9.5128
*
m-
m",m'
m /t , m'
m'
m'
150
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
TABLE LVI Continued.
Logarithmic. w w a
Unit- 1 radian.
Cos
*,
-*
M
*
fl
w'
-4r+ 4
7. 7640
7. 8364,,
m
~3/ 1 + 4
7. 4203
8. 3915
-2r+ 4
7. 8104 n
8. 6268
- r+ 4
8. 0479 n
8. 8018
4
4. 518 B
[5. 886 n ]
[5. 70 B ]
r+ 4
7. 1339
7.8500
2F+ 4
7. 8421
8. 4293,,
q r* _i_ A
O-* i~ **o
7. 9669
8. 6796 n
4r+ J
7. 9760
8. 7576
I' 2
-4r+40 +24 ,:'..:
6.9002
7. 6938 B
-3r +400+24
7. 1638
7. 8502 n
-2r+40 +24
- r+40 +24
8. I860,,
8. 4016
40o+24
3.76
6. 0608 n
8. 4157
8. 9760 n
9. 1661
+ r +40 +24
9. 1714
0. 1382 n
+2r+40 +24
8. 9358
9. 8333 B
+3r+40 +2J
8. 7718
9. 6681 B
+4r+40 +2J
8. 6236
9. 5372 n
l'*
3.76
5. 7516
4.7
r
7. 8677
8. 6727 B
2P
7. 8610 B
8. 2228
sr
8. 1026 B
8. 7296
ft 1 1 1 .9
4r
8. 1538 B
8. 8728
f
r
*Httt .0
7. 9418 B
8. 7337
zr
7. 9312
8.7154
sr
7. 7920,,
8. 6154
4r
7. 639 n
8.5001
^
?
-4r+40 +34-^o
Kit >
7.446
8. 1156 B
-3r+40 +3J -|o
W J
7. 1858
7. 8677 B
I r+40o+34-^o
7. 6176 B
7. 9693
40 +3J 0^*0
: i<HX) 1
4.804 n
7.168
7. 9368,,
8. 3724
r+40o+34--^o
UK h
7. 7887
8. 8492,,
2r +400+34 -2
ir.\ii> o
7.448
8. 4531,,
3/^+400+3 J ^o
<.'IH<', ,
7. 1976
8. 2026 n
4r+40 +34-^ (>
6.978
7. 9963 n
V
20 +24
5. 4181^
6.292
7. 4754 n
8. 6636
- v , 60 +64
5. 418 B
6.292
8. 6328 n
9. 4351
i)oV
20 + 4
5.885
6. 719 B
8.5059
9. 2804 B
20 +34
4.974
5. 896 B
8. 0326 B
8. 1975
60o+54
5.935
6. 780 B
9. 2774
0. 0330 n
% ,/s
20
5.744,
6.535
[8. 3811 n ]
9. 1030
5.44 B
6.327
8. 0917
8.6300
60o+44
5. 919 n
6.744
9. 4432 n
0.1464
i?"
20 + 4
5.301
6. 149 B
8. 2302
9. 0152 B
60o+34
5.301
6. 149 n
9.1294
9. 7729 n
/**
20o+24
8.5904
9. 3492 n
9.8022
20 + 4-^o
4. 502 B
5.41
8. 1011 n
8. 8726
4. 502 B
5.41
8. 0554 n
8. 9263
JV
20 + 4
8. 5592 B
9. 3245
200+24-^0
4.057
5. 021 n
6.887
8. 1804 B
60o+44-^o
4.057
5.021 n
8. 2718
9.1021.
m"
m n
-^
m -x
m'
m'
ww =2Cw*i)P'Tflj 1 t cos Agr., where C represents the respective coefficient.
No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
151
Turning now to the determination of e and K, let equations (235), (236) be written in the
form (244), where
1 2 , 1 1
"~+'V-
Multiplying the first of these by sin ^, the second by cos </> and adding,
S sin ^ + C cos <f> = g (
sn
coa<f>+z sin
cos <f>+z sin <l>)+-r z(y sin $ z cos ^)+ . . .
4C0
Here, again, the arguments and factors are functions of the elements a, e, JT, e, and the expansion
in a Taylor's series is necessary.
Let
S sin <{> + (7 cos <f>=f(i), F u U J, 2")
Then the form of Taylor's series is the same as the expression for w w , (p. 148), with the
following modification. Within first order quantities,
; .;
-fi
Hence,
, F lt 6 lt A, S) = n(y cos ^ + 2 sm
1.
= ^ (y sin s 2 cos e)
"
sn
'o w -
(1 TJ n COS
l- cos
The order of computation is : calculation of
A-SC{.. KU
1, . ,.
-s(y cos ^+2sm^)
by inspection of the table for W, in which the arguments are to be given the subscript zero,
differentiation of the first order terms, calculation of the necessary products of functions of y, z,
and the partial derivatives, and the addition of these products to the first calculation. The
required function is given in Table LVIL
e v-t- \ ---y
It 07
f.8f .S ,87 S
n t'
0"
|).':T> ?
>:
152
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
[Vol. XIV.
tufj .'ii ft-d l; fW
Logarithmic
TABLE LVII.
Ssin i/r+C cos <]>
j 7; on
Unit-1".
Cos
,-
UJ-J
UM
to"
w
*
^- 5 r+20 +24,
8.81
1. 082,
1. 5710
1. 612,
4.T+200+24]
9.009
1. 2314,
1.5492
0. 989 n
^ 3/ 1 +20 +24,
9.318
0.931
1.604,
1.916
4> 2.T+200+24,
<p- r +200+24,
9.207
9.711
[1. 6478]
1.950
2. 1070 n
2. 3426,
2. 2333
2. 3713
^ +20 +24,
9.196
2. 1712,
2. 5678
2. 565 n
4>+ r+200+24,
9. 230 n
2. 3541,
3. 1493
3. 7107 n
0+2/"+20 +24)
9. 220,
1. 9114 n
2. 6867
3. 1657,
tl>-\-3r-\-20 -\-2d
9. 724,
1. 5372,
2. 3831
2. 8623,
^+4r+20 +24,
9. 494,
1. 2544,
2. 1315
2. 6333,
^+5r+20 +24,
9.100 n
1.018,
1.9034
2. 4348,
ii
^_5r +40 +44,
ro wioiJ->ni
ft. 771,
1.042,
1.868
2. 357 B
^-3F+40 +44^
0.06V
0. 3185,
1. 723,
2. 1626,
2. 3515
2. 6961
2. 6814,
2. 9214,
dJ iliur
t- r+40+44^
<!> +40 +4J
r + r +400+44,
0+2r+40 +44,
9.199
9.04,
0. 497,
1. 0286,
[2. 6172]
0. 7226
0.669
[2. 7787,]
3. 2379,
[3. 2511 n ]
3. 1702
2. 7877
[3. 0649]
3. 1223
[3. 4930]
4. 1580 n
3. 7083,
3. 0993,
3. 9385,
4. 9365
4.3605
0+3/ 1 +40 +44,
0.9435
2. 5117
3. 4261,
4.0450
^+4.T+40o+44>
0. 5122
2. 2732
3. 2042,
fe
V>-5r
U'i \0 ', - ^.
9. 814 B
1.925
2. 634,
2.984
<j> 4r
-
0. 0434,
2. 0527
2. 6896,
2.9432
ip3r
0. 3541,
2.145
2. 675,
2.744
*ijizr
9.140
0. 362,
2. 1351
2. 3850,
2. 4864,
if, r
0.4164,
2.3504,
3. 0929
3. 5397,
<l>
9. 274 n
0. 1436,
y+ f
0. 3102,
2.497
3. 1875,
3. 5978
<l>-\-2r
9. 137,
9.918
1. 9006 n
1. 0453
2. 8834
<P+zr
9.465
0. 812,
2. 5218,
3.3564
^+4r
9.20
1. 406 n
1.729
J)'
V--5r+40 +34)
9.476
1.327
1. 889,
2. 2299
y 4/ 1 +40 +34>
9.781
1.447
2. 1506,
2.5419
^ 3/'+40 +34>
9.811
2. 1070
2. 6309,
2. 8608
y~ 2/*+40 +3J
0. 3489
2. 5095
2. 9557,
3. 0952
$ -f +40 +3 J
0. 9511
3. 3599
2. 7758
3. 9726
^ +40 +34(
8.76,
[0. 158]
[2. 7932 n ]
3. 3085
3. 4526,
^+ F +40 +34>
9.961,
3. 3609,
4. 3114
5. 0691 n
^+2f+40 +34i
0.491,
2. 9943,
3. 8728
4. 4922,
y+3/^+40 +34(
1. 0464,
2. 7293,
3. 6067
4. 1945,
^+4r+40 +3J
0. 678 B
2. 4992,
3. 3946
...mfc
<!> 4r+4i
J 'HB ftfl-tj
inyift ->iJi
9.848
0.0792
2. 0766,
2. 1609,
2.712
2. 6968
2. 9697,
2. 7976,
,t ,v to *t
^l2r+4
Ifl'-'-'fVJIT 91
9.013,
0.3941
0.248
2. 157,
2. 0455
2.491
2. 7898 n
1.51
3. 2380
'til 1 .f)'>
^ ^"+4i
r r 'I'tXi'ltj
9.901
2.584
3. 2539,
3. 6434
^ +4i
9. 885 B
0. 8518
v^+ ^"+4t
0.1664
1.836
2.448
3. 3029 B
^+2r+4,
9.009
9.76 B
2. 1633
2. 6170,
2. 2433
^+3r+4,
9.38,
2.1064
2. 7194,
2. 9212
^+4r+4,
1. 9892
2. 6870,
V
^-5r+60 +64,
2.3144
2. 9730,
^ 4r+60 +64i
2. 9538
3. 3785,
<j> 3/"+60 +64i
3. 3102
3. 5843,
V^ 2r+60 +64i
[3. 4970]
[3. 8423 n ]
Y *^+60o+64,
3. 9455
3. 7269,
<!> +60 +6J
9.95,
1. 1109,
3. 1673 B
[3. 9296]
[4. 3377 n ]
if>-\- .T+600+64)
3. 9144,
5. 0372
^+2/^+600+640
3. 5594,
4. 5942
^+ 3 r+60 +64,
3. 3121,
4. 3236
No. 3.]
Logarithmic
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
TABUS LVII Continued.
S sin <{>+C coo<!>
153
Unit-l"
COS K-
^
u-t
V*
W 1C*
V
^-5r+20 +2J.
2.1657
2. 7221 B
A 4/ I +20 +2 J
2.1255
2.8004,
^ 3r+20 +2J
2.234
3. 1304 n
it 2.F+200+2 J
2.576
3.3804,
^- r+20 +2j
3.1995
3.8325,
V> +20 +2J
0.344,
1.017
2. 689,
[3. 4822]
3.9938,
,5+ r+20 +2J,
2.2480
3.2839,
^+2r+20 +2J
9.45
3.1612
3.8424,
V
<i-5r-20 -2J.
2.700,
3.5481
^-4T-20o-2J
2. 817,
3.6251
^ &r 20 2J
2. 9247,
3.6905
<l> 2F 20 2 J
9.59,
3.0241,
3.7470
j>- r-20 -2J
3.1364,
3.8346
A -200-2J.
0.117
0.95,
2.297,
[2. 7856,]
3.6614
^+ T -20 -2 J,
2.8942,
3.5604
#+2r-20 -2J
2. 297,
3.1129
fci'
<f-5r+60 +5Jo
2.4885,
3.1691
^ 4/ 1 +60 +5 Jo
2. 976,
3.5560
^ 3/ l +60 ~t~5Jo
3.6541,
3.8829
^ 2f +60 +5J
[3. 9514,]
[4 1632]
^ /"+60o~t~5Jo
4.3903,
40037,
^ +60 +5J
0.295
L366
3.6364
[4. 3301,]
[46662]
i5+ /"+60 +5Jo
4.4005
5.4966,
c5+2/ I +60 +5J
4.0582
5.0612,
#+3r+60 +5J
3.8204
4.8027,
fcf*
^-5T+2o+ Jo
2-426,
3.0684
^ 4/"+20 + J
2.399,
3.0310
ci 3/ 1 +20o+ Jo
2.410,
3.1305
^ 2r*+20 + Jo
2.701,
3.4602
^ /^+20o+ Jo
3.2842,
3.8558
^ +20 + J0
0.444
1.188,
3.0569
[3. 7266,]
41122
<J>-\- ^"+20o+ Jo
2.8541
3.5823,
^+2r+20 + J
3. 2191,
3.7635
% ,/
^-5r-20 - J
3.1551
3.9530,
#-4r-20 - J
3.2454
3.3100
3.9948,
40023,
A 2/ 1 20 Jo
9.93
3.3277
3.9401,
^- r-20 - J
3. 1976
3.4598,
^ -20 - J
0.490,
1.324
3.0145,
3. 7326
42787
^+ /" 20 Jo
3.3632
3.9402,
<!>+ r-20 - J e
2.7792
3.5224,
loV
^-5r+20 +3J
2.2738,
2.847
<p 4r"+20 +3Jo
2.116,
3.0290
J> 3/^+200+3 J
2.5858,
3. 3787
A 2/"+20 +3 J
2.809,
3.5429
A- r+20 +3J
2.650,
3. 7297
J + r+20o+3J
9.98
0.60,
2.873,
[2.685]
3. 5126,
3.7980
4.2856
^+2r+20 +3J
9.46,
3.3438,
41208
q'i
s 5-5r+60o+4J
L9950
2.7422,
A 4r+60 +4J
2.6112
3.1949,
</> 3/"+60 +4 J
3.0556
3.5583,
y2f +60 +4 J
3.7934
3. 7947,
A /'+60 +4J
42260
44064
^ +60 +4J
9.98,
0. 76,
3. 5017,
41098
43552,
ifi-\- r +60 +4J
4.2852,
5. 3521
^+2r+60 +4J
3.9567,
49249
,1
^-5r+20 +2J
2.5018
3.0963,
</> 4/^+200+2 J
2.453
3.0935,
^ 3/^+200+2 J
2.4799
3. 2779,
A-W +20 +2 J
2.9375
3. 6294,
6- r+20 +2J
3.2833
3.8982,
f +2e o +2J
0.025,
0.60
2.634
3. 2781
4. 0439,
</>-\- /^+2^o+2Jo
3.5607
4 2381,
#+ 2 r+20.+2J.
3. 4629
4. 1704 n
154
Logarithmic
MEMOIRS NATIONAL ACADEMY OF SCIENCES.
TABLE L VI I Continued.
8 sin <j>+ C cos <}>
[Vol. XIV.
tJnit-l".
Cos
-.
^
-,
-
w
v>*
T)"
cj-5r-20
3.0090 B
3. 7477
^-4r-20o
3. 0676 B
3. 7445
<l>sr 20
3. 0764 n
3. 6664
^-2r-20
2. 958 n
3. 3121
^- r-20
3. 1140
4. 0201 B
<l> -20
0.305
1. 127 n
2.912
3. 5491 B
3.9085
^+ r-20
3. 0396 B
3. 6320
^+2r-20
1
2. 4706 B
3. 2330
f
^-5r+60 +54-.?o
2.006
2. 7505 n
<l>4r +60 +54 ^o
2.335
2. 981 B
<t> Zr +60 +54 2o
j
2.544
3. 1436 B
^ 2.T+600+54 (>
2.718
3. 2445 B
v ^*+60Q+5^o *o
2.970
2. 9116 n
dt +600+5.^0 ^*o
8.6 B
9.7
2. 114^
2.923
3. 4067 n
d>-}- F +60 +54 ^o
2. 7948 B
3.9420
^-j_2r+60 +54 2
2. 3824,
3.4488
P
^-5r +20 +24
9.6
2.387
<j> 4r+20 +24
1. 916 B
2.911
V ? irv
<!> 3r+20 +24
2. 5178 B
3.3047
J> 2J rl +20 +24
2. 938 B
3.6294
4,- r+20 +24
3. 3406 B
3. 9330
<{, +200+24
0. 5910
3. 1266
3. 8021 B
4.1894
^+ r+20 +24
frit<> .};
3. 4070
4. 3178 B
0+2r+20 +24
3.0472
3. 9308 B
f
0_5/--20 - 4+^o
0. 732 n
1.085
d>4r26 4+-^o
0.35
1. 895 n
<{izr20 4+^0
1.463
2. 5146 n
\t ,,jr
</r-2r-20 - 4+^o
2.064
3. 0255 n
^- r-20 - 4+^0
2. 6816
3. 6280 n
-20 - 4+^0
9.04
0.11 B
2.636
3. 3284 B
3. 7399
^+ r-20 - 4+J
3. 0572 B
3.6430
V-+2r-20 - 4+^o
2. 9121 n
3.5491
lo 8
4, +40 +44
0.775
1.66,
3. 1052 B
3. 0342 B
J +80^+84
0.29 B
0.65
1.10
1.54 n
3. 1888
3. 7520
3. 6104 n
4. 5812 B
V*
Vl'
V> +40 +54
3. 7577
4. 3244 n
<!> +40 +34
1.260 B
2.081
3. 1240
4. 1388
d> -40 -34
1.005
1.77 B
3. 5356 B
3. 3560
<!> +80 +7J
1. 228 B
2.093
4. 3980 n
5. 1827
lo V
^ +400+44
4. 1155 B
4.5547
+400+24
1.106
1.88 B
2.831
4. 1803 B
<!> -40 -24
1. 146 B
1.88
3.0422
4. 0180
^ +800+64
1.321
2. 152 B
4. 5658
5. 3010 n
if*
<j> +400+34
3. 8375
4. 0446 B
<l> -40 - 4
1 f'- !
iiii 'i
3. 2197
3. 9650 B
<[> +80 +5J
4. 2553 n
4. 9349
jj,
d, +40 +34-.r
3.0024
3. 8634 n
df -400-34+^0
9. 98 B
0.8
2. 956 B
3. 8331
J, -j.g^o+74 ^"0
3. 0757
3. 9759 B
,'{.' -.
d, +40 +44
0.46 n
1.32
3. 8514 W
4.6436
}' *>'
d> +40 +44-JT
2.442
1.846 B
<l> -40 -24+J
3. 2486
4. 0585 B
:
d, +80 +64-l-
3. 2818 n
4. 1441
* V'S'H '
<!> +40 +34
0.27
1. 15 n
3.9421
4. 6972 n
S sin <p-\-C cos (/'=SC l 1 u'*7jP7/9j 2 ' cos Arg, where C^ represents the coefficient.
No. 3.]
MINOR PLANETS LEUSCHNEE, GLANCY, LEVY.
155
COMPARISON OF TABLES.
Table LVI. Unless there are errors of calculation, all the discrepancies are due to the
accumulation of other discrepancies already discussed. Without going into the details of the
construction, it is sufficient to remark that our table is built from practically all of the available
auxiliary material. Our table includes many more terms than v. Zeipel's table, but it is wanting
in the two arguments 6F and SF in the first block of terms. These arguments contain 3s and 4e,
respectively, and our series were not inclusive of these higher multiples. It would be more
consistent to include them, since the argument 7F is included.
Table LVTI. Unless there are errors of calculation, all the discrepancies are due to the
accumulation of discrepancies already discussed. Our table is built from practically all the
available auxiliary material. Large disagreements are to be explained by v. Zeipel's use
of the formula following Z 131, equation (244). In this equation the following functions are
omitted:
cos
sn -
cos
sn
ERRATA 1 IN H. v. ZEIPEL, ANGENAHERTE JUPITERSTORUNGEN FtJR DIE HECDBA-GRUPPE.
With the exception of $ 6, Stdrungen des Radius-vector, all the developments have been checked.
Page.
Line.'
For
Read
dQ
1
8a
dx
&x
dQ
dQ
1
3ff
9a
dj
J
dv
/Ho
~dT
5
9
5b
2a
(15)
iW
(16)
3U
9
Ib
W+v 3
W+r*
12
2a
w
fY
W
12
9a
a+
n+i
P n+i
12
lOa
0*
/3 ()
'
12
6b
(2n+4t+l)V B (4i+4)-y < '
(2n+4i+l)Y < l -+(4t+4)7^
13
5a
(2n+4f+3) T< '-(4i+4)- x ^ 1
(2n+4t+l) 7< -+(4t+4)7'-*
14
2b
n'g>
ng'
noo
15
8bff
I
_ (j
B
16
lOa
sin
COS
16
6b
1 dQ
7 a *3I
1 dQ
19
Ga
dF
dF_
da a
<fa
20
5b
TJ-?
T**
21
4a
^.ro.n
1?j 3 -
21
4b
Metoden
Methoden
24
9b
2P ' .
2pi-. a
27
21a
P . fn 1. n+ljj,
/Vofn-?.-n+lJ_i
34
21a
P jtfi \ 7l + l)_j
42
44
5b
18b
/V 2 (n.-n)
.ffi.,(n+l.-n+l)
Fl'^n.-n)
45
46
20a
8a
g(n
l G (n
46
lOa
I. O (TI ! n+2) i
O ..(n ! Ti+2) t
46
lla
,. (n 1- n)+t
O .i(nl-n)-t
1 Inclusive of those tabulated by v. Zeipel.
' The number of the line counting from the top of the page is indicated by a, counting from the bottom of the page by b.
On page 3 and all following pages ' is denned by ' The error consists in the omission of a statement announcing a change of notation.
See definition of >' given on page 2.
156 MEMOIRS NATIONAL ACADEMY OF SCIENCES.
Errata in H. v. Zeipel, Angenahcrte Jupittrstarungen fur die Hecuba-Gruppe Continued.
[Vol. XI V_
Page.
Line.'
For
Read
J>Q
dfl
49
7b
f^
Os
dr
50
6b
r 2
a 2
r ''iimytxi owj orii a
a 2
50
6b
3+ij 2
3+14, 2
51
lb
5 .,(n.-n+l)
53
lib
>;
>"
54
5a
E
-i
56
4b
61
lib
20+20
20+24
62
17a
^+60+40
^-)-60+4J
62
5b
+436
+439
63
65
9a
3a
f(l-ecos)TF' 2 ]
(106)
t(i-3oo)F / ]
(106a)
65
5a
(106)
(106a)
68
3a
W * W~ 3 tti~ J W~ l V)
tu~ 4 w"" 3 tt>~" 2 it*" 1 w t^
69
6b
sin A
77P7j /C! j 2 'sin A
69
69
5b
4b
sin (A <l>+t)
sin (A-\-(f>c)
Tjpr] vj sin ^vi y-pj
7)P73 9?^ sin ^^4 -4-(A ^
70
lb
W t '"
w
70
lb
cos A
TfPl)'9f* COS ^1
71
7a
ess A
-rjp-rfqft cos ^.
75
15a
4o-
4o-2
75
18a
4o-i
75
2b
4H
75
lb
AI-O
79
lOb
e
^<
81
8b
1ecoB c)
(1 e cos )
83
12a
+3744
+3344
86
4a
(128j) und (130)
(1282), (128 3 ) uod (130)
86
6a
o W^
Ot?
T^ 1
91
9a
e COB*
e cos
91
lla
TP 2
ffj'
92
3a
1J, COS C
y, COS
92
lOb
-i(l-e cos ) (IF-Jff) (W+iB)
-J[(l- cos ) (W-IS) (W+iS)]
92
4b
^
^
93
lOa
sin A
yPij'Qj 2 * gin A
93
lOa
2'
J
94
19b
dW
dF
97
15a
(156)
(154)
99
4b
T)W sin t)
s '
100
5a
A'
^
100
6a
Ju^
_lj5
115
116
4b
7a
8SJ> -I
(192)
116
lOb
/o ro i\
~-j- ^Oj l^lj/
119
(i)
4-?
f7p.
122
3a
123
4b
(i f 2 f)
(1 f 2 f 8 )
125
3a
1 o COS 2
1 COS
128
7a
A
4ft
129
5a
fi
P
131
7b
(0 jl )
131
6b
$+A+l)
(^_)-^4_ )
132
8a
2.9227 B
1.9227 n
132
26a
5.3376
5.0376
134
9a
9H4Q
(4t>+/ 4 )
135
lOa
w
2
T
135
lla
to
2
I
140
141
26a
6a
[nSz]
0*8998
Mel.
0/8998
i Th number of the line counting from the top of the page Is indicated by a, counting from the bottom of the page by b.
'No. 3.]
MINOR PLANETS LEUSCHNER, GLANCY, LEVY.
157
ERRATA IN KARL BOHLIN. SDR LE DfiVELOPPEMENT DBS PERTURBATIONS PLANETAIRES, 1-7,
AND TABLES I-XX.
Page.
Line.-
For Read
3
5a
/( , =(1+m)a ,
-.
/.'=(! +m)a
dW
dW
11
Ib
14
lla
14V
l-r j
20
3a
+i s cos 2*
-i^ COB 2
29
8b
r*
y'-n
29
2b
V I/'
e -V-i/'
30
lla
a
?
a'
?
30
lla
e'
X?
+p
30
12a
2n+m-l
2n+m+l
30
13a
/?T\*
f.-. n wi-
+'. ^jg -r g
30
13a
2n+7n 1
--!.- . l-.'v.nW
2n+m+l
30
13a
2n+wi-2
2n+7n+2
30
14a
2n+m-l
2n-i-"i-|-l . '
30
5b
e V-i(*-O
e ^in( r ^)
33
9a
r*2i+ */_, i _, \
10 \ * "/
Xjf*U 1. )
35
4a
(73)
(74)
36
la
2/V'"e'^~ 1 <- T >
"*|+|Y
2f 4 * " V IO
36
lOa
^.,(n-l.+n+l)
^..(n-l.-n+l)
38
13a
a
a
38
lOb
e J^lr- w>)
38
3b
2(ij')y'- 1
2/ . /\|_i
38
2b
2n)y'- 1
2(jj)y'-l
40
3a
K (n. 1 n)
JP / n 1 ji\
41
13a
a
VO
a'"
45
2a
iT . (0.-n)
i" . (n. n)
45
9b
a
4A
7n
r(r 1)V~'
" : |."'-
, T(rl)*x f ~ l
^O
i
1.1.2
h 1.1.2
46
7b
(n-)(n-t+2)
,'V-
(n *)(n +2) n _f^
2
2 ' r ^
46
5b
(n-3)
(n )
46
4b
v
n' 4
48
14a
P'jK/n 2. n)
P',. (n-2.-n)
48
7b
pi . \n-\-\. n 2]
P*,.- ITI-)-!. n 2!
48
7b
pi (n+1 n 2
pi (n+1 n 2)
48
6b
P^'.iin-l.-n-Zi
P',.,|n 1. n 2|
50
5a
R 1 .~(n-^-l.n\
_ T
/fi -(n+1. n I)!/
50
9b
R l \,o(nl.n+l
+1-
R l \ < ,(n\.n-\-\)+tf
50
3b
Rig-fn. n+2^+1*
R l (n n+2)+ /
51
Ib
P'(n+r.-n+* i
piln-f-r. n+^
59
5a
fi 3 ' 1 , jn+l. n]
^J3-l _Tjj 1. 71]
59
8a
^ 3>l o-i[ n -~ n +l]
J-l e r n __ B ^-i]
60
6a
if
/
60
8a
(See footnote z )
ftf\
QK
("^/t )
/N
~T~\7n.ftf
DV
VO
24
V^
24
60
fib
jX""**)
( +)
61
7b
P .,[n.+n+l]
P .i[".-n+l]
61
5b
Poit n -+ n -lJ
P In n 11
62
la
n P-
n /i
6
6
62
7a
P,., (n+2.-n-l)
P 2 .,(n+2.-n+l)
62
7a
^2^
2
62
63
8a
5a
pJ.'o(n-l'.-n+l)H
-
P.Jn+l.-n+l]
P . (n-l.-n+l)_*
63
63
63
63
lla
13a
14a
5b
Pj'o M-L-n-1]-!
PO-O n-l.-n-lj-j
RO-O n. n+lj-r'
P 1 . (n+2.-n-l)+,
P . c fn-l.-n+l]_*
P . n-l.-n+l]_*
P.o-0 n.-n IJ-r'
63
2b
RO-O i. n 1] i*
63
Ib
R . n._n !]-'
R . n. n 1]-^
1 The number of the line countin? from the top of the pase is indicated by a. counting from the bottom of the page by b.
i The argument a is defined first by eq. (31), p. 20, secondly by eq. (105), p. 60. The first of these definitions is used in J 8.
158 MEMOIRS NATIONAL ACADEMY OF SCIENCES.
Errata in Karl Bohlin, Sur le Developpement des Perturbations Planitaires, 1-7, and Tables I-XX Continued
Page.
Line."
For
Read
64
64
6a
12a
.Ro-o[".--l]-*'
R 2 . [n.-n+l]+ x '
64
64
66
14a
15a
7a
(n-n)
#,.,[7+l. -]_
F (n+r. -n+s
&r T "*
Ftn+r.-n+t)
66
8a
G (n+r. n+s
G (n+r. -n+s)
+3
+3
70
la
-3 P,. 2 (n+l.-n-2)
-2 Pj. 2 (n+l.-n-2)
2
-2
71
4a
F t (n. n+l)+
F t a ( n _ w _)-i) + .
+3
+3
71
9b
-2 P lf0 (n.-n+l)-3
-2 P,. (n.-n+l)_*
2
-2
73
2a
F 2 . u (n.n+l)- x '
.F 2 . (n.-w-l)-,r'
73
18a, ff.
See foot note. 2
f
73
4b
R . (n. n+l)+ x
R . (n.-n+l)+ x >
73
74
3b
8a
R . n.-n+l)+ x >
R . (n.-n+I)+ x '
75
75
la
lib '
jOn.-n+l)-,'
G . (n+l.-n)+,+j
78
Ib
To 1 '"
Ti 1 '* 1
79
Ib
fjTn+i.n
y, m-f2.n
79
*)
n=l
n
80
9b
T(+i m ' n
^ <+j m.n
81
12a
3
81
13a
(120)
(120)*)
135
7a
-^D-3 1 '*
jf i.
139
3a
/y**
2ri'-
140
2a
/Y -n
2r 1 '-
154
la
(86)
(93)
161
la
-or
or
169
8b
3
I
a
170
3a
2S 2r<3< *
2T<'-
170
4b
^
i 2 ^ 3>B
171 ff.
See foot note.'
185
2a
3. 27886
3. 27887
185
13b
4
3
188
6b
2. 017 3 n
2. 01703,,
189
14a
3. 27886
3. 27887
197
16a
0. 146128 B
1. 146128 n
197
18b
1. 505151
1. 505150
198
15b
1. 662759 n
1. 662758 n
198
2b
0. 477121
0. 477121 n
1 The number of the line counting from the top of the page is indicated by a, counting from the bottom of the page by b.
1 The space between lines 18 and 19 should read }'.
' Tables XII, XIII. XIV give the same coefficients in numbers as Tables XVI, XVII, XVIII give in logarithms, respectively. The same factor
should therefore occur in the former.
o
!-"